EP2186469B1 - Method and device for the localisation of fluorophores or absorbers in surrounding medium - Google Patents

Description

TECHNICAL FIELD AND PRIOR ART

The invention relates to the field of diffuse optical imaging on biological tissues by time-resolved methods.

It applies to the localization of fluorophores or absorbers in a diffusing medium such as, for example, an organ of an animal or a human being (brain, breast, any organ where fluorophores can be injected).

In particular, an interest of a molecule that absorbs is as follows. Some cancerous tumors are visible by the fact that they have a greater attenuation coefficient than healthy tissues. In this case, it is important to be able to identify these tumors.

In this context, it may also be necessary to use a fluorophore, which is a specific fluorescent marker. It is preferentially attached to the target cells of interest (for example cancer cells). Such a fluorophore offers a better contrast of detection than a non-specific marker. Fluorescence molecular imaging optical techniques are aimed at spatially locating these fluorescent markers.

Optical tomography systems use a variety of light sources. There is therefore devices that operate in continuous mode, others in frequency mode (which use frequency modulated lasers) and finally devices operating in time mode, which use pulsed lasers.

Thus, there are three methods of diffuse optical imaging, which differ depending on whether the light source used is continuous, frequency (i.e. whose intensity is modulated by a sinusoidal function of time) or pulsed.

Instruments using a continuous light source were the first to be used, the light source being a filtered white source or a monochromatic source, such as a laser. Point or two-dimensional detectors are then used, measuring the intensity of the light reflected or transmitted by a fabric illuminated by the light source.

The second category of diffuse optical imaging, called frequency imaging, uses a light source modulated in intensity at a given frequency. The light source is most often a laser source intensity modulated at frequencies f generally between a few tens of kHz to a few hundred MHz. The detector used measures both the amplitude of the light signal reflected or transmitted by the tissue, as well as the phase of this light signal relative to that of the light source.

Finally, the third category of diffuse optical imaging is pulsed optical diffuse imaging, also known as diffuse optical imaging. impulse, or temporal, or resolved in time. The source used produces pulses of pulses, that is to say of short duration, at a given repetition rate. The sources used may be pulsed picosecond laser diodes, or femtosecond lasers. The duration of the pulse being generally less than 1ns, then we speak of subnanosecond pulsed light sources. The repetition rate is most often between a few hundred kHz to a few hundred MHz.

The invention relates to this third category of fluorescence imaging.

Document WO2006 / 032151 discloses a method and an optical imaging system for the localization of fluorophores or absorbers in a tissue where a time spreading function is measured in several different pairs of positions of the source and the detector, moments of these curves are calculated and used to determine the position of said fluorophore or absorber.

Time data is the one that contains the most informational content on the fabric that is traversed, but for which reconstruction techniques are the most complex. The measurement at each acquisition point is indeed a time dependent function (called TPSF for "Temporal Point Spread Function", or time spreading function).

It seeks to extract simple parameters of the TPSF, which is also known theoretical expression. Then, the resolution of the inverse problem makes it possible to find the distribution of the fluorescent markers.

In some cases, it is sought to locate a fluorophore for later intervention. At present, the location of diseased cells can be done using biopsy techniques. But, in order to find the position, even approximate, of these cells, one must make several samples. This invasive technique is obviously difficult to implement and is time consuming.

There is therefore the problem of determining the location, even approximately, of a fluorophore or an absorber fixed on an area of a medium, for example a biological medium, with a limited number of measurements, and a relatively low stripping time, even close to real time. The invention finds all its utility for making such a determination in media with limited accessibility, for example when screening for prostate cancer tumors from endoscopic measurements, or when screening for breast cancer, or osteosarchomia ....

STATEMENT OF THE INVENTION

• un signal de fluorescence (Φ_fluo) émis par ce fluorophore,
• ou un signal réémis à la même longueur d'onde que celle d'excitation (issue de la source de rayonnement) par l'élément absorbant qui absorbe et diffuse le rayonnement absorbé. L'élément absorbant peut faire partie d'une zone d'un milieu qui présente un coefficient d'absorption supérieur à celui du milieu diffusant. On a déjà donné l'exemple de certaines tumeurs cancéreuses.

The invention firstly relates to a method for locating at least one fluorophore or at least one element (in particular a molecule) that is absorbent - also called an absorber - in a diffusing medium, using a suitable radiation source. to emit excitation radiation from this fluorophore and detection means able to measure:

• a fluorescence signal (Φ _fluo ) emitted by this fluorophore,
• or a signal re-emitted at the same wavelength as that of excitation (from the radiation source) by the absorbing element which absorbs and diffuses the absorbed radiation. The absorbent element may be part of an area of a medium that has an absorption coefficient greater than that of the medium broadcasting. We have already given the example of certain cancerous tumors.

1. a) pour au moins 3 couples différents de positions de la source de rayonnement et des moyens de détection, au moins une excitation par un rayonnement provenant de la source de rayonnement, et au moins une détection par les moyens de détection du signal de fluorescence émis par ce fluorophore après cette excitation, ou du signal d'émission émis par l'absorbeur ou par l'élément absorbant,
2. b) pour chacun de ces couples, l'identification d'une surface sur laquelle le fluorophore (ou l'élément absorbant) est situé, ou d'un volume comprenant cette surface et dans lequel le fluorophore ou l'absorbeur est situé,
3. c) une estimation de la localisation du fluorophore ou de la localisation de l'élément absorbant dans son milieu diffusant, par calcul de l'intersection des trois surfaces, ou éventuellement dans un volume autour de cette intersection.

This locating method comprises the following steps:

1. a) for at least 3 different pairs of positions of the radiation source and detection means, at least one excitation by radiation from the radiation source, and at least one detection by the fluorescence signal detection means by this fluorophore after this excitation, or of the emission signal emitted by the absorber or by the absorbing element,
2. (b) for each of these pairs, the identification of a surface on which the fluorophore (or absorbent element) is located, or a volume comprising that surface and in which the fluorophore or absorber is located,
3. c) an estimation of the location of the fluorophore or the location of the absorbent element in its scattering medium, by calculating the intersection of the three surfaces, or possibly in a volume around this intersection.

The signals can be detected for example by TCSPC technique or by camera.

Such a method can be implemented regardless of the respective positions of the fluorophore or absorber, the source and the detector.

The accuracy of the method will not be the same depending on whether one takes positions of the source and detection means near or far from the fluorophore or absorber. But the process allows in all cases a location, even rough, with a small number of measurements, with a fast stripping, even in real time. Such a method therefore makes it possible to direct the performance of biopsies or complementary non-invasive examinations.

In addition, this method is very quick to implement, it is possible to locate a fluorophore or absorber in less than one minute, or even in less than 15 seconds.

For each pair of at least 3 different pairs, it is possible to define a histogram representing the number of photons detected as a function of time; this histogram can be called the fluorescence time function.

The source of radiation or light and / or the detection means may be respectively the end of an optical fiber which brings the excitation light into the medium and / or the end of an optical fiber which takes a part emission light. For the source, it can also be a light source of the Laser or LED type. Regarding the detector, it may be a photomultiplier tube, or an image sensor.

According to the invention the element targeted by the detection may be a single fluorophore, or a plurality or an aggregation of fluorophores grouped substantially in one place. It can also be an absorber or a plurality of absorbers grouped substantially in one place (aggregation). In the rest of the text, an absorber will be assimilated to a fluorophore whose emission wavelength is equal to the excitation wavelength, and whose lifetime is zero. In other words, an absorbing medium, or an absorber, is a fluorophore whose emission wavelength is equal to the excitation wavelength, and whose time constant τ is zero. Only the term "fluorophore" will be used, except when specificities of the absorbers are explained.

The volume around the intersection of the three calculated surfaces can result from the intersection of three volumes, each volume being associated with one of the surfaces and comprising all the points of the surface and all the points which satisfy the surface equation with a margin of error, for example ± 10% or ± 5%.

The diffusing medium surrounding the fluorophore may have any shape of geometry: for example, it is of infinite type.

It can also be of semi-infinite type, limited by a wall. A semi-infinite medium, for which each of the fiber ends of the excitation signal and the fluorescence signal collection signal is disposed more than 1 cm, or 1.5 cm or 2 cm from the limiting wall. this medium can be treated, from the point of view of the process according to the invention, as an infinite medium.

The diffusing medium can still form a slab (geometry called "slab"), limited by two parallel surfaces. The invention also applies to a medium of any shape, the outer surface of which is decomposed into a series of planes.

In some embodiments, the surfaces at the intersection of which the fluorophore is located may be in the form of ellipsoids, but may also be 3D surfaces not having this shape.

According to one embodiment, step b) comprises, for each position pair, a calculation of a normed time moment, of order n (n 1 1), of a fluorescence time function. Such a moment can be obtained by an n-order derivative of a transform of said fluorescence time function, for example a Fourier transform, or a Laplace transform or a Mellin transform.

We recall that a normed moment of order n (n≥ 1), denoted m _n of a distribution g (t) is defined by $$m_{\mathrm{not}}=\frac{\underset{-\infty }{\overset{+\infty }{\int }}{t}^{\mathrm{not}}\mathrm{boy\; Wut}\left(t\right)\mathit{dt}}{\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{boy\; Wut}\left(t\right)\mathit{dt}}$$

In some cases, it may be advantageous to carry out excitation by a light source, and detection by the detection means, for 4 different pairs of positions of the radiation source and detection means.

• sélectionner parmi les quatre couples, le couple de points, dit quatrième couple de points, pour lequel la durée moyenne d'arrivée des photons, de l'émetteur au récepteur, est la plus courte, et calculer, pour ce quatrième couple de points, l'équation d'une surface sur laquelle le fluorophore est situé ;
• et, pour chacun des trois autres couples de points, après calcul d'une première équation d'une surface sur laquelle le fluorophore est situé, on peut effectuer une différence entre cette équation et celle associée au quatrième couple, pour obtenir une deuxième équation indépendante de la durée de vie du fluorophore.

We can then:

• selecting from the four pairs, the pair of points, said fourth pair of points, for which the average duration of arrival of the photons, from the transmitter to the receiver, is the shortest, and calculate, for this fourth pair of points, the equation of a surface on which the fluorophore is located;
• and, for each of the other three pairs of points, after calculating a first equation of a surface on which the fluorophore is located, a difference can be made between this equation and that associated with the fourth pair, to obtain a second independent equation the life of the fluorophore.

According to one embodiment, it is therefore possible, during step b) for each pair of positions, to calculate an equation of each surface, independent of the lifetime of the fluorophore, which can result from the difference between the calculated average time for each position torque and the calculated average time for another position torque. This other pair of positions can be the one having a minimum average time or having a minimum calculated average time.

A method according to the invention may further comprise a visual or graphical representation of the position or distribution of the fluorophore or absorbers.

The invention also relates to a method for locating an area in a medium, comprising introducing into this medium at least one fluorophore or an absorber, and the localization of this fluorophore or absorber in said medium by a method according to the invention. The zone may be for example a medium consisting of a human or animal organ, the fluorophore or absorber for marking this zoned. The goal may be to locate diseased cells in such a medium. As already indicated above, the invention makes it possible to achieve this location in a fast time, or even in real time.

A method according to the invention can be implemented invasively or non-invasively.

The invention also relates to a device for implementing a method according to the invention, and for locating a fluorophore in a diffusing medium.

• au moins une source de lumière ou de rayonnement, apte à émettre un rayonnement d'excitation de ce fluorophore,
• au moins un moyen de détection apte à mesurer un signal de fluorescence (Φ_fluo) émis par ce fluorophore,
• des moyens de calcul, programmés pour calculer :
  1. a) pour chaque couple de points parmi au moins 3 couples différents de positions d'une source de rayonnement et d'un moyen de détection, la détermination, après une excitation réalisée par un rayonnement provenant de la source de rayonnement, et une détection par les moyens de détection des signaux de fluorescence émis par ce fluorophore après cette excitation, d'une surface sur laquelle le fluorophore est situé, ou un volume comprenant cette surface et dans lequel le fluorophore est situé,
  2. b) une estimation de la localisation du fluorophore dans son milieu environnant, par calcul de l'intersection des trois surfaces ou d'un volume autour de cette intersection.

The invention therefore also relates to a device for locating a fluorophore in a scattering medium, which may be of one of the types mentioned above, comprising:

• at least one light or radiation source capable of emitting excitation radiation from this fluorophore,
• at least one detection means capable of measuring a fluorescence signal (Φ _fluo ) emitted by this fluorophore,
• calculating means, programmed to calculate:
  1. a) for each pair of points among at least 3 different pairs of positions of a radiation source and detection means, the determination, after an excitation performed by radiation from the radiation source, and a detection by the means for detecting the fluorescence signals emitted by this fluorophore after this excitation, of a surface on which the fluorophore is located, or a volume comprising this surface and in which the fluorophore is located,
  2. b) an estimate of the location of the fluorophore in its surrounding environment, by calculating the intersection of the three surfaces or a volume around this intersection.

The surrounding medium may be of infinite type or semi-infinite type or shaped slice, limited by two parallel surfaces, or any shape, its outer surface being decomposed into a series of planes.

The calculation means, for example electronic means such as a microcomputer or a computer or a microprocessor, are programmed to store and process fluorescence or emission data obtained from the detection means. Preferably these means are programmed to implement a treatment method according to the invention.

The calculation means make it possible to calculate, for each position pair, a normed time moment, of order n (n 1 1), of a fluorescence time function. This moment can be obtained from a frequency function deduced from said fluorescence time function, for example by Fourier transform or by Mellin transform.

According to an advantageous embodiment, the calculation means allow or are programmed to perform, for the localization of at least one fluorophore or absorber, and for each position pair, a calculation of an equation of each surface, independent of the lifetime of the fluorophore. For this purpose, the calculation means make it possible to make the difference between the average time calculated for each position torque and the average time measured or calculated for another position torque.

The device allows excitation by the radiation source, and detection by the detection means for 4 different pairs of positions of the radiation source and detection means.

The fourth position torque may be a pair for which the fluorescence signal has a measured average time or a calculated average time less than that of the 3 position pairs.

BRIEF DESCRIPTION OF THE DRAWINGS

• The Figures 1A and 1B each represent an example of an experimental device for implementing the invention,
• the Figures 2A and 2B represent respectively a series of laser pulses and single emitted photons, and a fluorescence curve obtained from the data relating to single photons,
• the figure 3 represents a theoretical configuration, in two dimensions,
• the Figures 4A-4F represent the experimental setup of a realized measurement, the positioning of the source and detection points, the measured fluorescence curves, and the calculated position and actual position of a fluorophore;
• the Figure 5A represents pairs of positions of a source and a detector, in an XY plane, for a measurement according to the invention, in semi-infinite configuration,
• the Figures 5B-5D represent portions of ellipsoids used in the context of a measurement according to the invention, in semi-infinite configuration,
• the Figures 6A-6C each represent a localization of a fluorophore obtained with a process according to the invention, in a semi-infinite configuration,
• the Figures 7A - 7D represent surfaces used in the context of another embodiment of the invention,
• the Figures 8A-8D represent portions of surfaces used in the context of a measurement according to the invention, in slice configuration.
• the figure 9 represents a flow diagram of a method according to the invention.
• the Figures 10A-10D represent various arrangements, with respect to a medium studied, excitation and detection means, or optical fibers leading to this medium excitation radiation and transmitting a radiation to be detected.

DETAILED PRESENTATION OF PARTICULAR EMBODIMENTS

The Figure 1A is an example of experimental system 2 for implementing a method according to the invention. Schematic representations of a medium and means of excitation and detection are given later in connection with the Figures 10A-10D .

The device of the Figure 1A uses as detector 4 a photomultiplier and a TCSPC card (Time Correlated Single Photon Counting), in fact integrated in a set of data processing means 24.

The light is emitted by a pulse source 8. The duration of each pulse is generally less than 1 ns, for example between between a few hundreds of fs, for example 500 fs, or 100 ps or 500 ps and second 100 ps or 500 ps or 1 ns. The repetition rate is preferably between a few hundred kHz, for example 100 kHz or 500 kHz and a few hundred MHz, for example 100 MHz or 500 MHz.

The light emitted by the source 8 is sent and collected by fibers 10, 12 which can be displaced. The two fibers can be mounted on means of displacement in vertical and horizontal translation (along the X, Y and Z axes of the Figure 1A ). For this purpose, it is possible to use one or more plates mounted in translation to move the ends of the optical fibers. One can also use a laser, also mounted on a plate mounted in translation. A detector can also be mounted on such a plate. In all cases the translation can be computer controlled.

The source 8 of radiation pulses can also be used as a means of triggering the TCSPC card (see link 9 between the source 8 and the means 24). It is also possible to work with pulses in the field of femto seconds, provided that the appropriate radiation source is available, that is to say a laser source 8, each pulse of which has a temporal width also in the domain of the second femto.

According to a particular embodiment, the source 8 is a pulsed laser diode, for example at a wavelength close to 630 nm and with a repetition rate of about 50 MHz.

The laser light preferably passes through an interference filter 14 to eliminate any secondary modes.

The fiber 12 collects light from the medium studied, in particular the light diffused by this medium. An interference filter 16 and / or a low wavelength absorbing color filter can be placed in front of the detector 4 to select the fluorescence light (for example: λ> 650 nm, the source being at a wavelength of, for example, 631 nm) of a fluorophore 22 disposed in the medium 20 and to optimize the removal of the excitation light.

In the text of the present application, whatever the embodiment of the invention, it will include question of the position of the radiation source, and / or the position of the detection means. When fibers are used, these positions are most often understood as those of the ends of the fibers which bring the radiation into the diffusing medium 20, or at the interface of this medium, and / or those that collect the scattered radiation, the latter being placed in the medium or at the interface of this medium, but do not then agree as being those the actual source 8 or the detector 4 itself.

As a variant, it is possible to use a light source delivering a light beam, for example a transmitting laser, and a camera provided with an objective lens, which allows optical coupling between the sensitive face of the camera and the face of the camera. exit from the observed object (which is the case of the device of the Figure 1B ).

In "endoscopic" type application, a fiber system is preferably used, the fibers being able for example to be integrated in an endo-rectal probe, at least one fiber transmitting the pulsed excitation light source to the zone to be examined. This pulsed light source may be external to the probe. At least one other fiber transmits the optical transmission signal of the area to be examined to a photodetector, which may also be external to the probe. According to the TCSPC (Time Correlated Single Photon Counting) technique, a photon emitted by the fluorophore is detected by means of the photomultiplier after a pulse of the radiation source.

The system therefore allows time-resolved detection of the fluorescence pulses. It allows to recover fluorescence photons.

The Figure 2A represents a series of laser pulses Li (i = 1 - 4) and a series of corresponding single photons pi (i = 1 - 4), the latter being detected by a TCSPC system (Time Correlated Single Photon Counting). Each photon is actually detected with respect to the start of the corresponding pulse: on the Figure 2A , ti represents the time elapsed between each laser pulse Li and the detection time of each photon pi.

It is then possible to establish a statistical distribution or histogram of the arrival time of the photons, as illustrated on the Figure 2B , which represents the number of fluorescence photons detected, as a function of the elapsed time t with respect to each laser pulse. From such a histogram, it is possible to determine statistical parameters, in particular an average of the arrival time, or average time (it is in fact the average of the abscissa weighted by the ordinates of the histogram).

Such a curve Φ _fluo (t) which, as we see (see also the example of the figure 4C ), makes it possible to use all the information over a large time window, on either side of the point of maximum intensity (and not only in the rising part of the signal) can then be processed to derive characteristic information , such as arrival time, or average time as will be explained below.

Electronic means 24 such as a microcomputer or a computer are programmed to store and process the data of the TCSPC card. More specifically, a central unit 26 is programmed to implement a processing method according to the invention. Display or display means 27 make it possible, after treatment, to represent the positioning of the fluorophore (s) in the medium under examination.

Other detection techniques may be employed, for example with an intensified camera, and for example a high-speed intensified camera of the "gated camera" type; in this case the camera opens on a time gate, of width for example about 200 ps, then this door is shifted, for example 25 ps in 25 ps.

The Figure 1B is an example of another experimental system 2 using as detector 32 a fast camera: the relative position of the source (or the detector) and the object can easily be realized.

A fluorescence excitation beam 20 of a medium 20, containing one or more fluorophores 22, is emitted by a radiation source 37 (not shown in the figure) which may be of the same type as that presented above. liaison with the Figure 1A . A photodetector 36 makes it possible to control means 40 forming a delay line. Reference 24 designates, as on the Figure 1A electronic data processing means of the microcomputer or computer type, programmed to store and process the data of the camera 32. A central unit of these means 24 is programmed to implement a processing method according to the invention. Here again, means of display or visualization allow, after treatment, to represent the positioning or the location of one or more fluorophores in the examined medium.

The distance between the light beam emitted by the source, and the detector, or the part of the medium optically coupled to the detector, is for example between a few mm and a few cm, for example between 1 mm and 20 cm in the case of biological media. . This is particularly the case in the case of integration into an endo-rectal probe. The relative positioning of the source and the detector with respect to an organ to be examined can be determined by means of the optical or ultrasonic type, for example a ultrasound probe.

• une position de source S1 (cas n = 1) successivement avec 3 positions distinctes D1, D2, D3 de détecteur (cas p=3) ; donc on utilise les couples de positions (S1, D1), (S1, D2), (S1, D3),
• deux positions distinctes S1 et S2 de la source (cas n = 2) avec 2 ou 3 positions distinctes de détecteur (cas p=2 ou 3) ; donc on utilise par exemple les couples de positions (S1, D1), (S2, D2) et (S1, D2) ou (S1, D1), (S2, D2) et (S1, D3),
• trois positions distinctes S1, S2 et S3 de la source (cas n = 3) avec 1 ou 2 positions distinctes de détecteur (cas p=1 ou 2) ; donc on utilise par exemple les couples de positions (S1, D1), (S2, D1) et (S3, D1) ou (S1, D1), (S2, D2) et (S3, D2).

As a variant of the above devices, it is possible to place a number n of light sources and a number p of detectors, with for example 1≤n≤3 and 1≤p≤3, the light sources and the detectors possibly being fiber ends. optical connected respectively to at least one pulsed light source or at least one photodetector, of the photomultiplier or camera type. These positions are then combined so as to have at least 3 distinct pairs of source positions and detector positions:

• a source position S1 (case n = 1) successively with 3 distinct detector positions D1, D2, D3 (case p = 3); therefore we use the pairs of positions (S1, D1), (S1, D2), (S1, D3),
• two distinct positions S1 and S2 of the source (case n = 2) with 2 or 3 distinct positions of detector (case p = 2 or 3); therefore, the pairs of positions (S1, D1), (S2, D2) and (S1, D2) or (S1, D1), (S2, D2) and (S1, D3) are used,
• three distinct positions S1, S2 and S3 of the source (case n = 3) with 1 or 2 distinct detector positions (case p = 1 or 2); therefore, the pairs of positions (S1, D1), (S2, D1) and (S3, D1) or (S1, D1), (S2, D2) and (S3, D2) are used, for example.

3 pairs are preferably used for which the 3 source positions are distinct from each other and the 3 detector positions are distinct from each other (configuration: (S1, D1), (S2, D2) and (S3, D3)).

The excitation light at the wavelength λ _x excites the fluorophore which re-emits a so-called emission light at the wavelength λ _m > λ _x with a lifetime τ. This lifetime corresponds to the average duration during which the excited electrons remain in this state before returning to their initial state. The following developments show that the case of an absorber, and therefore of a wavelength emission substantially equal to that of excitation with a zero lifetime, can be deduced from the case of a fluorophore.

où

• r_s désigne la position de la source,
• r désigne une position quelconque dans le milieu,
• la fonction Φ_x(r, t) désigne le débit de fluence du rayonnement d'excitation au point r à l'instant t.
• le coefficient D_x(r) est le coefficient de diffusion au point r, à la longueur d'onde d'excitation λ_x ;
• S_x(r _s, t) désigne le flux de rayonnement émis, à l'instant t, par la source au point r_s.

The equation of propagation of the excitation light in a diffusing medium produced by the pulsed source of radiation is written: $$\frac{1}{{\mathrm{vs}}_{\mathrm{not}}}\frac{\partial {\mathrm{\phi }}_x\left(\mathbf{r},t\right)}{\partial t}+\tilde{\nabla }\mathrm{.}\left(-D_x\left(\mathbf{r}\right)\nabla {\mathrm{\phi }}_x\left(\mathbf{r},t\right)\right)+{\mu }_{\mathit{ax}}\left(\mathbf{r}\right){\mathrm{\phi }}_x\left(\mathbf{r},t\right)=S_x\left({\mathbf{r}}_s,t\right)$$

or

• r _s denotes the position of the source,
• r denotes any position in the medium,
• the function Φ _x ( r, t) designates the flow rate of the excitation radiation at point r at time t.
• the coefficient D _x (r) is the diffusion coefficient at the point r, at the excitation wavelength λ _x ;
• S _x ( r _s , t) denotes the radiation flux emitted at time t by the source at the point r _s .

avec :

• la fonction Φ_m(r, t) qui désigne le débit de fluence du rayonnement d'émission au point r, à l'instant t ;
• le coefficient D_m(r), qui est le coefficient de diffusion au point r à la longueur d'onde d'émission λm.

That of the emission light is written: $$\frac{1}{{\mathrm{vs}}_{\mathrm{not}}}\frac{\partial {\mathrm{\phi }}_m\left(\mathbf{r},t\right)}{\partial t}+\tilde{\nabla }\mathrm{.}\left(-D_m\left(\mathbf{r}\right)\nabla {\mathrm{\phi }}_m\left(\mathbf{r},t\right)\right)+{\mu }_{\mathit{am}}\left(\mathbf{r}\right){\mathrm{\phi }}_m\left(\mathbf{r},t\right)=\frac{1}{\tau }\mathrm{\eta }{\mu }_{\mathit{ax}-->m}{\int }_0^t\exp \left(\frac{-\left(t-\mathit{t\; \text{'}}\right)}{\tau }\right){\mathrm{\phi }}_x\left(\mathbf{r},\mathit{t\; \text{'}}\right)\mathit{dt\text{'}}$$

with:

• the function Φ _m ( r, t) which designates the fluence rate of the emission radiation at the point r, at time t;
• the coefficient D _m (r), which is the diffusion coefficient at the point r at the emission wavelength λm.

$${\mathrm{\phi }}_m\left(\mathbf{r},t\right)=2{\mathit{AD}}_m\left(\mathbf{r}\right)\frac{\partial {\mathrm{\phi }}_m}{\partial n}\left(\mathbf{r},t\right)=0$$

et ce pour tout point r de la frontière δΩ du milieu diffusant.The boundary conditions being as follows: $${\mathrm{\phi }}_x\left(\mathbf{r},t\right)=2{\mathit{AD}}_x\left(\mathbf{r}\right)\frac{\partial {\mathrm{\phi }}_x}{\partial \mathrm{not}}\left(\mathbf{r},t\right)=0$$

$${\mathrm{\phi }}_m\left(\mathbf{r},t\right)=2{\mathit{AD}}_m\left(\mathbf{r}\right)\frac{\partial {\mathrm{\phi }}_m}{\partial \mathrm{not}}\left(\mathbf{r},t\right)=0$$

and this for every point r of the δΩ boundary of the scattering medium.

These equations are also developed and explained in the thesis of Aurélie Laidevant, "Optical methods resolved in time for fluorescence tomography in diffusing media", Joseph Fourier University, Grenoble, October 5, 2006, p.51 and following.

• µ_ax et µ_am sont les coefficients d'absorption du milieu diffusant à la longueur d'onde d'excitation λ_x et d'émission λ_m respectivement,
• µ_ax→m est le coefficient d'absorption associé à la concentration locale de fluorophores à la longueur d'onde d'excitation λ_x;
• η est le rendement quantique du fluorophore,
• Φ(r,t) est le débit de fluence associé à la densité de photons, en photons/m²s, ou, de manière équivalente, en Watt/m²,
• A est égal à (1-R_eff)/(1+R_eff), où R_eff est le coefficient de réflexion dû à la différence d'indice entre l'air et le milieu 22 environnant le fluorophore.

In these equations:

• μ _ax and μ _am are the absorption coefficients of the scattering medium at the excitation wavelength λ _x and emission λ _m respectively,
• μ _{ax → m} is the absorption coefficient associated with the local concentration of fluorophores at the excitation wavelength λ _x ;
• η is the quantum yield of the fluorophore,
• Φ (r, t) is the fluence rate associated with the photon density, in photons / m ² s, or, equivalently, in Watt / m ² ,
• A is equal to (1-R _eff ) / (1 + R _eff ), where R _eff is the reflection coefficient due to the difference in index between the air and the medium 22 surrounding the fluorophore.

The boundary condition expresses that, at the middle boundary, the diffused fluence rate is zero.

The theoretical function Φ _m can be considered as proportional to the TPSF, the latter being measured experimentally ( Figure 2B ). Thus, the normalized time moments of the TPSF are equal to the normalized time moments of the function Φ _m .

The resolution of the system of coupled equations obtained is not easy, and the exploitation of the volume of data contained in the time signal is difficult. We are generally interested in the exploitation the time interval extracted from these time-dependent functions (called TPSF for "Temporal Spread Function", or time-stretching function) - to perform a fitting step with the experimental data.

To exploit the temporal information by means of other quantities extracted from the measured TPSF, other measures can be defined such as the measurement of the Mellin transform times of the TPSF, or the Fourier Transforms (and thus to work in frequency mode), or Laplace Transforms or by parameterization of the TPSF (that is to say the determination of characteristic parameters such as the coordinates of the maximum, and / or the coefficient of rising slope and / or downward slope of the interest of such quantities is to reduce the volume of data by removing any redundancies existing between them

In the remainder of the invention, the main interest is in calculating n-order moments of the histogram, such a calculation being possible by Fourier transform.

Le moment d'ordre 0 est: $$m_o=\underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{\phi }}_m\left(r,t\right)\mathit{dt}$$

In particular, from the function Φ _m , it is possible to calculate any normalized time moment, and in particular the normed first order moment. Such a magnitude corresponds to the mean arrival time of the photons. The normalized time moments of the function Φ _m , are defined as: $$m_k=\underset{-\infty }{\overset{+\infty }{\int }}{t}^k{\mathrm{\phi }}_m\left(r,t\right)\mathit{dt}/\underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{\phi }}_m\left(r,t\right)\mathit{dt}$$

The moment of order 0 is: $$m_o=\underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{\phi }}_m\left(r,t\right)\mathit{dt}$$

où <t>_x, <t>_m et <t>_{déclin fluo} désignent respectivement la durée moyenne du trajet source-fluorophore, la durée du trajet fluorophore -détecteur, et la durée de vie moyenne de l'espèce excitée (cette dernière étant égale à 0 en présence d'un absorbeur).On rappelle que, en outre :

• r_sr = |r_s - r|
• r_rd = |r-r_d|
• m₁ est le résultat de la mesure

We will now treat the case of an infinite medium. We can show that the normed time-time of order 1 of the function Φ _m , defining the average time of arrival of the photons <t> ∞, in infinite medium is written: $$\begin{array}{r}{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">m</figure-callout>}}_1=\underset{-\infty }{\overset{+\infty }{\int }}t{\mathrm{\phi }}_m\left(r,t\right)\mathit{dt}/\underset{-\infty }{\overset{+\infty }{\int }}{\mathrm{\phi }}_m\left(r,t\right)\mathit{dt}={<t>}_{\infty }={<t>}_x+{<t>}_m+{<t>}_{\mathit{fluorescent\; decline}}\\ =\frac{{\mathrm{<figure-callout\; id="2"\; label="r\; sr"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">r\; </figure-callout>}}_{\mathit{sr}}}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{ax}}{D}_x}}+\frac{{\mathrm{<figure-callout\; id="2"\; label="r\; rd"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">r\; </figure-callout>}}_{\mathit{rd}}}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{am}}{D}_m}}+\mathrm{\tau }\end{array}$$

where <t> _x , <t> _m and <t> _{fluo decay} are the average duration of the fluorophore source path, the fluorophore-detector trip duration, and the average duration of the excited species (the latter being equal to 0 in the presence of an absorber). It is recalled that, in addition:

• r _sr = | r _s - r |
• r _rd = | rr _d |
• m ₁ is the result of measurement

We see that we have here the equation of a 3D surface, here having the shape of an ellipsoid.

Où Φ̃(ω) est la transformée de Fourier de Φ_m: $$\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)=\frac{1}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt}\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}=\frac{-i}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}t\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt}$$

More generally, the normed moment of order k can be expressed in the form: $$m_k={\left(i\right)}^k\frac{{\partial }^k\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial {\mathrm{\omega }}^k}{|}_{\mathrm{\omega }=0}\times \frac{1}{\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)|\mathrm{\omega }=0}$$

Where Φ (ω) is the Fourier transform of Φ _m : $$\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)=\frac{1}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt}\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}=\frac{-\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">i</figure-callout>}}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}t\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt}$$

soit en milieu infini $V=\frac{r_{\mathit{sr}}}{2{c_n}^2\sqrt{{\mathrm{\mu }}_{\mathit{ax}}^2{D}_x}}+\frac{r_{\mathit{rd}}}{2c_n\sqrt{{\mathrm{\mu }}_{\mathit{am}}^2{D}_m}}+{\mathrm{\tau }}^2\mathrm{.}$

For example, we can show that the variance of the temporal distribution is written $V=<{\left(t-<t>\right)}^2>=m_2-{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">m</figure-callout>}}_1^2,$

either in infinite medium $V=\frac{{\mathrm{<figure-callout\; id="2"\; label="r\; sr"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">r\; </figure-callout>}}_{\mathit{sr}}}{2{{\mathrm{vs}}_{\mathrm{not}}}^2\sqrt{{\mathrm{\mu }}_{\mathit{<figure-callout\; id="2"\; label="ax"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">ax</figure-callout>}}^2{D}_x}}+\frac{{\mathrm{<figure-callout\; id="2"\; label="r\; rd"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">r\; </figure-callout>}}_{\mathit{rd}}}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{am}}^2{D}_m}}+{\mathrm{<figure-callout\; id="2"\; label="\tau "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^2\mathrm{.}$

It is observed that we obtain, also in the case of variance, the definition of a three-dimensional surface, here that of an ellipsoid.

This magnitude (the variance) can be calculated in the context of the present invention from moments of order 2 and 1.

More generally still it is possible, according to the invention, to calculate any moment of order greater than 1, and to establish a relationship linking at least one of these moments to the respective distances source-fluorophore and fluorophore-detector. If one considers moments of higher order, the equation connecting them, or a combination of these, to the respective distances source-fluorophore and fluorophore-detector, is also that of a three-dimensional surface.

Preferably, the standardized first order moment is used.

In general, a fluorophore can be seen to be able to absorb a certain amount of light (in its excited state) and to re-emit a portion of it at a higher wavelength (when it is de-excited), with a delay corresponding to the duration in the excited state. From the expressions given above, it is therefore possible to pass from the case of the fluorophore to the single absorber by writing μ _ax = μ _am , D _x = D _m and τ = 0. This applies to all the cases described in the present application. The above considerations therefore also apply to the case of an absorbent object.

The infinite medium is typically the one encountered when invasive measurements are made, with fibers 10, 12 for bringing light into a medium 21 (case of the figure 10A ) and to take the fluorescence signal emitted by the fluorophore F, the ends of the fibers being positioned in the diffusing medium, each at a distance d ₁₀ and d ₁₂ of at least 1 cm or 1.5 cm from a wall 21 which delimits the medium. We can more generally consider that we are in an infinite medium when the source and the detector are placed at a sufficient depth in the scattering medium. In practice, this corresponds to invasive measurements, made for example by placing optical fibers, or a portion of these fibers, inside a diffusing medium.

We consider a first hypothesis, that of a medium in which the optical properties are identical to the two excitation and measurement wavelengths, λ _x and at λ _m , which is for example substantially the case when these two wavelengths are separated by for example 10 nm or a few tens of nm. Recall also that λx λm when one has a non-fluorescent absorber. ; The optical properties of the medium at both wavelengths can therefore be considered identical. We then have a simplification of equation (7) above: $$r_{\mathit{sr}}+r_{\mathit{rd}}=2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathrm{at}}D}\left({<t>}_{\infty }-\mathrm{\tau }\right)$$

multiplié par le temps moyen d'arrivée des photons corrigé du retard introduit par le temps de vie τ.This is the formula of an ellipsoid of foci S and D, the location of the points r such that the source-fluorophore distance r _sr plus the fluorophore-detector distance r _rd is equal to the product of the apparent velocity in the diffusing medium $2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathrm{at}}D}$

multiplied by the mean arrival time of the photons corrected for the delay introduced by the life time τ.

Thus, to find the 3D position of the fluorophore, it suffices to perform 3 measurements at 3 source positions and 3 detector positions. The fluorophore is at the intersection of 3 ellipsoids defined in space by 3 different equations. Preferably, couples of positions defining directions substantially perpendicular to each other are selected. For each pair, we proceed to the establishment of a TPSF or a histogram as explained above.

In the second hypothesis, the case where the optical properties can not be considered identical to the 2 wavelengths, equation (7) becomes: $$\begin{array}{|l|}\hline \begin{array}{l}\frac{r_{\mathit{sr}}}{{\mathrm{\nu }}_x}+\frac{r_{\mathit{rd}}}{{\mathrm{\nu }}_x}={<t>}_x+{<t>}_m=\left({<t>}_{\infty }-\mathrm{\tau }\right)\\ {\mathrm{\nu }}_{x,m}=2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{ax},m}{D}_{x,m}}\end{array}\\ \hline\end{array}$$

This is always the equation of an ellipsoid but it now translates the location of the points r such that the source-fluorophore distance r _sr is "weighted" by the apparent velocity v _x (hence the term r _sr / v _x ) , and the fluorophore-detector distance r _rd is "weighted" by the apparent velocity v _m (hence the term r _rd / v _m ).

Again, to find the 3D position of the fluorophore, 3 measurements are made at three source positions and 3 detector positions. The fluorophore is at the intersection of 3 ellipsoids defined in space by 3 different equations. Preferably, couples of positions defining directions substantially perpendicular to each other are selected. For each pair of positions (source, detector), one proceeds to the establishment of a TPSF or a histogram as explained above. Also preferably, during each acquisition, we will try to increase as much as possible the source-detector distance.

The mean time expressions can be analytically given for geometries other than those of the infinite medium, more complex: semi-infinite medium, with a plane interface; middle plane to parallel faces ...

où $$\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)=\frac{1}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt},\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}=\frac{-i}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}t\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt}$$

In general, if we know the expression, analytic or numerical, of the Fourier transform Φ (ω) of the function φ _m , then: $${\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">m</figure-callout>}}_1=<t>=i\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}|\mathrm{\omega }=0\times \frac{1}{\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)|\mathrm{\omega }=0},$$

or $$\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)=\frac{1}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt},\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}=\frac{-\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">i</figure-callout>}}{\sqrt{2\mathrm{\pi }}}\underset{-\infty }{\overset{+\infty }{\int }}t\mathrm{\phi }\left(t\right)e^{-i\mathrm{\omega }t}\mathit{dt}$$

The described surface is no longer an ellipsoid, but remains a 3D surface; So, again, a fluorophore is located by looking for intersections between 3 surfaces, each obtained for a different point pair (source position, measurement position).

In general, in all the cases concerned by the invention (those above and the others exposed below), to find the 3D position of the fluorophore, three measurements are made with three positions, preferably different between they, from sources and 3 positions, preferably different from each other, detectors (in other words, a measurement for each of the three pairs (r _si , r _di ), i = 1,2,3, with r _si ≠ r _sj , for i ≠ j ≠ r and r _{di dj} for i ≠ j). The fluorophore is at the intersection of 3 ellipsoids defined in space by 3 different equations. Preferably, three pairs of positions defining directions substantially perpendicular to each other are selected. In some cases, of which we will see an example later, it may be useful to perform 4 measurements for 4 pairs of positions (source, detector). For each pair of positions (source, detector), proceeds to the establishment of a TPSF or a histogram as explained above.

Another example is the case of a semi-infinite medium. This is the case of any body probed on the surface, for example prostate, breast, brain, whose thickness is sufficiently large in front of the characteristic distances source-detector.

This case is the one encountered when measurements are made, with fibers 10, 12 for bringing light and for taking the fluorescence signal, the ends of the fibers being positioned on the surface of a wall 21 which delimits the middle ( figure 10B ). More generally, this is the case where source and detector are positioned on the surface of this wall 21 which defines the diffusing medium in which the fluorophore F is disposed.

This is particularly the case where one wishes to be non-invasive, source and detector being positioned near the surface of the medium. Thus, in the case of a diagnosis of a prostate tumor, source and detector will be placed on the periphery of an endorectal probe.

pour tout point r de la frontière δΩ du milieu diffusant, et cela pour les longueurs d'onde d'excitation et d'émission.In this case, it is necessary to check not only the propagation equation but also the associated boundary conditions, already indicated above: $$\mathrm{\phi }\left(\mathbf{r},t\right)=2\mathit{AD}\left(\mathbf{r}\right)\frac{\partial \mathrm{\phi }}{\partial \mathrm{not}}\left(\mathbf{r},t\right)=0,$$

for any point r of the δΩ boundary of the scattering medium, and this for the excitation and emission wavelengths.

We show that this condition is equivalent to writing, more simply, _extrapolated φ ( r , t ) = 0 ∀ r ∈ ∂Ω, where the extrapolated boundary is located at an _extrapolated distance = 2AD from the physical boundary, as explained in the article of A. Laidevant et al. "Effects of the surface boundary on the determination of the optical properties of a turbid medium with time resolved reflectance", Applied Optics, 45, 19, 4756 (2006) ).

• on connait la solution en milieu infini pour une source placée en r+. Une façon de satisfaire φ(r,t) =0 ∀r∈ ∂Ω _extrapolée est de placer une source virtuelle, de contribution négative, à une position r-, symétrique de r+ par rapport à la frontière extrapolée ∂Ω _extrapolée .

One way to satisfy this condition is to use the image source method, as mentioned in the article above, but also as explained in the thesis Aurélie Laidevant, "Optical methods resolved in time for fluorescence tomography in diffusing media", Joseph Fourier University, Grenoble, October 5, 2006, p.55-56 :

• we know the solution in infinite medium for a source placed in r +. One way to satisfy _extrapolated φ ( r , t ) = 0 ∀ r ∈ ∂Ω is to place a virtual source, of negative contribution, at a r- position, symmetrical of r + with respect to the extrapolated _extrapolated boundary ∂Ω.

avec : $$k^2\left(\mathrm{\omega }\right)=-\frac{{\mathrm{\mu }}_a}{D}+i\frac{\mathrm{\omega }}{c_{n}D}$$

et :

• Φ̃^∞+ : transformée de Fourier du débit de fluence émis par la source positive dans l'hypothèse d'un milieu infini
• Φ̃^∞- : transformée de Fourier du débit de fluence émis par la source négative dans l'hypothèse d'un milieu infini r+ = |r_s ⁺ - r| distance à la source excitatrice positive, la source excitatrice positive étant la source excitatrice
• r- =|r_s ^- - r|: distance à la source excitatrice négative, cette source négative étant factice, et définie afin d'obtenir φ(r,t)=0 ∀r∈ ∂Ω _extrapolée
• r_s+ : coordonnées de la source excitatrice positive
• r_s- : coordonnées de la source excitatrice négative G ^∞(k(ω),r) : fonction de Green satisfaisant l'équation de diffusion en milieu infini. $G^{\infty }\left(k\left(\mathrm{\omega }\right),r\right)=\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right)r}}{r}$
  

And all this is valid also in the frequency domain, and we have: $${\tilde{\mathrm{\Phi }}}^{1/2\infty }\left(\mathrm{\omega }\right)={\tilde{\mathrm{\Phi }}}^{\infty +}\left(\mathrm{\omega }\right)-{\tilde{\mathrm{\Phi }}}^{\infty -}\left(\mathrm{\omega }\right)=\frac{1}{4\mathrm{\pi }{\mathrm{vs}}_{\mathrm{not}}D}\left[\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right)r_{+}}}{r_{+}}-\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right)r_{-}}}{r_{-}}\right]=\frac{1}{4\mathrm{\pi }{\mathrm{vs}}_{\mathrm{not}}D}\left[{\mathrm{BOY\; WUT}}^{\infty }\left(k\left(\mathrm{\omega }\right),r_{+}\right)-{\mathrm{BOY\; WUT}}^{\infty }\left(k\left(\mathrm{\omega }\right),r_{-}\right)\right]$$

with: $${\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}}^2\left(\mathrm{\omega }\right)=-\frac{{\mathrm{\mu }}_{\mathrm{at}}}{D}+i\frac{\mathrm{\omega }}{{\mathrm{vs}}_{\mathrm{not}}D}$$

and

• Φ ^{∞ +} : Fourier transform of the fluence rate emitted by the positive source in the hypothesis of an infinite medium
• Φ ^∞- : Fourier transform of the fluence rate emitted by the negative source in the hypothesis of an infinite medium r + = | r _s ⁺ - r | distance to the positive excitatory source, the positive excitatory source being the excitatory source
• r- = | r _s ^- - r |: distance to the negative excitation source, this negative source being dummy, and defined in order to obtain φ ( r , t ) = 0 ∀ r ∈ ∂Ω _extrapolated
• r _{s +} : coordinates of the positive excitatory source
• r _s- : coordinates of the negative exciter source G ^∞ ( k (ω), r ): function of Green satisfying the diffusion equation in infinite medium. ${\mathrm{BOY\; WUT}}^{\infty }\left(k\left(\mathrm{\omega }\right),r\right)=\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right)r}}{r}$
  

On the other hand, we saw that ${\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">m</figure-callout>}}_1=<t>=i\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}|\mathrm{\omega }=0\times \frac{1}{\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)|\mathrm{\omega }=0}$

We deduce, for the semi-infinite medium: $$\begin{array}{l}{<t>}_{x,1/2\infty }=\frac{1}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{ax}}{D}_x}}\frac{e^{\mathit{ik}\left(0\right)r_{+}}-{\mathrm{<figure-callout\; id="0"\; label="e\; ik"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">e\; </figure-callout>}}^{\mathit{ik}}}{\frac{{\mathrm{<figure-callout\; id="0"\; label="e\; ik"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">e\; </figure-callout>}}^{\mathit{ik}}}{r_{+}}-\frac{{\mathrm{<figure-callout\; id="0"\; label="e\; ik"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">e\; </figure-callout>}}^{\mathit{ik}}}{r_{-}}}=\frac{1}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{ax}}{D}_x}}\frac{e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_x}}{D_x}}\left|{\mathbf{r}}_{\mathbf{s}+}-\mathbf{r}\right|}-e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_x}}{D_x}}\left|{\mathbf{r}}_{\mathbf{s}-}-\mathbf{r}\right|}}{\frac{e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_x}}{D_x}}\left|{\mathbf{r}}_{\mathbf{s}+}-\mathbf{r}\right|}}{\left|{\mathbf{r}}_{\mathbf{s}+}-\mathbf{r}\right|}-\frac{e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_x}}{D_x}}\left|{\mathbf{r}}_{\mathbf{s}-}-\mathbf{r}\right|}}{\left|{\mathbf{r}}_{\mathbf{s}-}-\mathbf{r}\right|}}=\frac{f\left(\left|{\mathbf{r}}_{\mathbf{s}+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}-}-\mathbf{r}\right|,k_x\left(0\right)\right)}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{ax}}{D}_x}}\\ \iff \\ {<t>}_{x,1/2\infty }=\frac{1}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{at}}D}}\frac{r_{+}{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{+}\right)-r_{-}{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{-}\right)}{{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{+}\right)-{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{-}\right)}\end{array}$$

Avec :

• r_d ⁺ : position de la source positive, qui est dans ce cas la source fluorescente
• r_d ^- : position de la source négative de la source fluorescente, source factice permettant d'obtenir φ _m (r ,t)=0 ∀r∈ ∂Ω _extrapolée
• r'+ = |r_d ⁺ - r|= distance à la source fluorescente positive
• r'- = |r_d ^- r|= distance à la source fluorescente négative

Similarly, we can show that $$\begin{array}{l}{<t>}_{m,1/2\infty }=\frac{1}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{am}}{D}_m}}\frac{e^{\mathit{ik}\left(0\right){\mathit{r\text{'}}}_{+}}-{\mathrm{<figure-callout\; id="0"\; label="e\; ik"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">e\; </figure-callout>}}^{\mathit{ik}}}{\frac{{\mathrm{<figure-callout\; id="0"\; label="e\; ik"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">e\; </figure-callout>}}^{\mathit{ik}}}{{\mathit{r\text{'}}}_{+}}-\frac{{\mathrm{<figure-callout\; id="0"\; label="e\; ik"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">e\; </figure-callout>}}^{\mathit{ik}}}{{\mathit{r\text{'}}}_{-}}}=\frac{1}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{am}}{D}_m}}\frac{e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_m}}{D_m}}\left|{\mathbf{r}}_{d+}-\mathbf{r}\right|}-e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_m}}{D_m}}\left|{\mathbf{r}}_{d-}-\mathbf{r}\right|}}{\frac{e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_m}}{D_m}}\left|{\mathbf{r}}_{d+}-\mathbf{r}\right|}}{\left|{\mathbf{r}}_{d+}-\mathbf{r}\right|}-\frac{e^{-\sqrt{\frac{{\mathrm{\mu }}_{{\mathit{at}}_m}}{D_m}}\left|{\mathbf{r}}_{d-}-\mathbf{r}\right|}}{\left|{\mathbf{r}}_{d-}-\mathbf{r}\right|}}=\frac{f\left(\left|{\mathbf{r}}_{d+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{d-}-\mathbf{r}\right|,{\mathrm{<figure-callout\; id="0"\; label="k\; m"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k\; </figure-callout>}}_m\left(0\right)\right)}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{am}}{D}_m}}\\ \iff \end{array}$$

With:

• r _d ⁺ : position of the positive source, which is in this case the fluorescent source
• r _d ^- : position of the negative source of the fluorescent source, dummy source making it possible to obtain φ _m ( r , t) = 0 ∀ r ∈ ∂Ω _extrapolated
• r '+ = | r _d ⁺ - r | = distance to the positive fluorescent source
• r'- | r _d ^- r | = distance to the negative fluorescent source

The use of an image source in semi-infinite medium is also described in the article of MSPaterson et al. "Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties", Applied Optics, vol.28, p.2331 - 2336, 1989 . This document also deals with a "slab" shaped geometry.

We can show that for the fluorescence signal, we always have: $$\begin{array}{|c|}\hline \begin{array}{l}{<t>}_{1/2\infty }={<t>}_{x,1/2\infty }+{<t>}_{m,1/2\infty }+\mathrm{\tau }\\ =\frac{\mathit{f}\left(\left|{\mathbf{r}}_{\mathbf{s}+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}-}-\mathbf{r}\right|,k_x\left(0\right)\right)}{{\mathrm{\nu }}_x}+\frac{\mathit{f}\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}-}\right|,{\mathrm{<figure-callout\; id="0"\; label="k\; m"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k\; </figure-callout>}}_m\left(0\right)\right)}{{\mathrm{\nu }}_m}+\mathrm{\tau }\end{array}\\ \hline\end{array}$$

This defines an implicit function and a surface in three-dimensional space.

Here again, and as already explained above, to find the 3D position of the fluorophore, 3 measurements are made at three source positions and 3 detector positions. The fluorophore is therefore at the intersection of 3 surfaces defined in space by 3 different equations. For each pair of positions, one proceeds to the establishment of a TPSF or a histogram as explained above.

Preferably, couples of positions defining directions substantially perpendicular to each other are selected.

In practice, the conditions defined by the geometry in a semi-infinite medium are not very different from those of the infinite medium, since the source and the detector are positioned (in the case of Figure 1A , the ends of the fibers 10 and 12) away from the interface, more than 1 cm or 2 cm from it (as in figure 10A ). In this case, indeed, everything happens as if the medium was infinite, from the point of view of the fluorophore.

en se plaçant dans le cas d'une géométrie infinie. Il n'y a pas de source fictive et on a donc :

• r_s+ = r_s,
• r- = |r_s- - r| - +∞,
• r'^- = |r - r_d-| = +∞,
• r_d+ = r_d

We can check the validity of the equation $$\begin{array}{|c|}\hline \begin{array}{l}{<t>}_{1/2\infty }={<t>}_{x,1/2\infty }+{<t>}_{m,1/2\infty }+\mathrm{\tau }\\ =\frac{\mathit{f}\left(\left|{\mathbf{r}}_{\mathbf{s}+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}-}-\mathbf{r}\right|,k_x\left(0\right)\right)}{{\mathrm{\nu }}_x}+\frac{\mathit{f}\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}-}\right|,{\mathrm{<figure-callout\; id="0"\; label="k\; m"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k\; </figure-callout>}}_m\left(0\right)\right)}{{\mathrm{\nu }}_m}+\mathrm{\tau }\end{array}\\ \hline\end{array}$$

placing himself in the case of an infinite geometry. There is no fictional source, so we have:

• r _{s +} = r _s ,
• r- = | r _s- - r | - + ∞,
• r ' ^- = | r - r _d- | = + ∞,
• r _{d +} = r _d

We then find the expression corresponding to the infinite medium.

We will now deal with the case of a medium with parallel faces (geometry in the form of "slice" or "slab"). This type of geometry is already discussed in the article by M.S.Paterson et al. already quoted above.

In this case, it is shown that for the excitation signal: $${{\tilde{\mathrm{\Phi }}}_x}^{\mathit{slab}}\left(\mathrm{\omega }\right)={\displaystyle \underset{i}{\Sigma }}\left[{{\tilde{\mathrm{\Phi }}}_x}^{\mathit{\infty }+}\left(r_{s+,i},\mathrm{\omega }\right)-{{\tilde{\mathrm{\Phi }}}_x}^{\mathit{\infty }-}\left(r_{s-,i},\mathrm{\omega }\right)\right]=\frac{1}{4\mathrm{\pi }{\mathrm{vs}}_{\mathrm{not}}{D}_x}{\displaystyle \underset{i}{\Sigma }}\left[\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right)r_{i+}}}{r_{i+}}-\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right)r_{i-}}}{r_{i-}}\right]$$

The summation in this expression results from the fact that we are in the presence of two parallel planes which limit the medium. There is therefore a contribution from two negative virtual sources, but each of the two planes multiplies the effects of the negative virtual source associated with the other plane. We must add the contribution of these different effects, in the form of a sum to infinity indexed by i.

Similarly, for the transmission signal: $${{\tilde{\mathrm{\Phi }}}_m}^{\mathit{slab}}\left(\mathrm{\omega }\right)={\displaystyle \underset{i}{\Sigma }}\left[{{\tilde{\mathrm{\Phi }}}_m}^{\mathit{\infty }+}\left({\mathit{r\text{'}}}_{+,i},\mathrm{\omega }\right)-{{\tilde{\mathrm{\Phi }}}_m}^{\mathit{\infty }-}\left({\mathit{r\text{'}}}_{-,i},\mathrm{\omega }\right)\right]=\frac{1}{4\mathrm{\pi }{\mathrm{vs}}_{\mathrm{not}}{D}_m}{\displaystyle \underset{i}{\Sigma }}\left[\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right){\mathit{r\text{'}}+}_i}}{{\mathit{r\text{'}}}_{+i}}-\frac{e^{\mathit{ik}\left(\mathrm{\omega }\right){\mathit{r\text{'}}-}_i}}{{\mathit{r\text{'}}}_{-i}}\right]$$

$${{<t>}_x}^{\mathit{slab}}=\frac{1}{2c_n\sqrt{{\mathrm{\mu }}_{{\mathit{a}}_x}{D}_x}}{\displaystyle \sum _i}\frac{r_{+,i}{G}^{\infty }\left(k\left(0\right),r_{+,i}\right)-r_{-,i}{G}^{\infty }\left(k\left(0\right),r_{-,i}\right)}{G^{\infty }\left(k\left(0\right),r_{+,i}\right)-G^{\infty }\left(k\left(0\right),r_{-,i}\right)}={\displaystyle \sum _i}\frac{f\left(\left|{\mathbf{r}}_{s+,i}-\mathbf{r}\right|,\left|{\mathbf{r}}_{s-,i}-\mathbf{r}\right|,k\left(0\right)\right)}{2c_n\sqrt{{\mathrm{\mu }}_{\mathit{ax}}{D}_x}}$$

et $${{<t>}_m}^{\mathit{slab}}=\frac{1}{2c_n\sqrt{{\mathrm{\mu }}_{{\mathit{a}}_m}{D}_m}}{\displaystyle \sum _i}\frac{{\mathit{rʹ}}_{+,i}{G}^{\infty }\left(k\left(0\right),{\mathit{rʹ}}_{+,i}\right)-{\mathit{rʹ}}_{-,i}{G}^{\infty }\left(k\left(0\right),{\mathit{rʹ}}_{-,i}\right)}{G^{\infty }\left(k\left(0\right),{\mathit{rʹ}}_{+,i}\right)-G^{\infty }\left(k\left(0\right),{\mathit{rʹ}}_{-,i}\right)}={\displaystyle \sum _i}\frac{f\left(\left|{\mathbf{r}}_{d+,i}-\mathbf{r}\right|,\left|{\mathbf{r}}_{d-,i}-\mathbf{r}\right|,k\left(0\right)\right)}{2c_n\sqrt{{\mathrm{\mu }}_{{\mathit{a}}_m}{D}_m}}$$

Or, $${<t>}^{\mathit{slab}}1/2\infty ={<t>}^{\mathit{slab}}x,1/2\infty +{<t>}^{\mathit{slab}}m,1/2\infty +\mathrm{\tau }$$

Soit : $$\begin{array}{|cc|}\hline {<t>}_{\mathit{slab}}={\displaystyle \sum _i}\left[\frac{\mathit{f}\left(\left|{\mathbf{r}}_{\mathbf{s}i+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}i-}-\mathbf{r}\right|,k_x\left(0\right)\right)}{{\mathrm{\nu }}_x}+\frac{\mathit{f}\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}i+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}i-}\right|,k_m\left(0\right)\right)}{{\mathrm{\nu }}_m}\right]+\mathrm{\tau }& \\ \hline\end{array}$$

We call <t> _slab the normalized first order moment of Φ ^slab ( t ) , which can be obtained by the Fourier transform Φ slab (ω) and ${\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">m</figure-callout>}}_1=<t>=i\frac{\partial \tilde{\mathrm{\Phi }}\mathit{slab}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}|\mathrm{\omega }=0\times \frac{1}{\tilde{\mathrm{\Phi }}\mathit{slab}\left(\mathrm{\omega }\right)|\mathrm{\omega }=0}:$

$${{<t>}_x}^{\mathit{slab}}=\frac{1}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{{\mathit{at}}_x}{D}_x}}{\displaystyle \underset{i}{\Sigma }}\frac{r_{+,i}{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{+,i}\right)-r_{-,i}{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{-,i}\right)}{{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{+,i}\right)-{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),r_{-,i}\right)}={\displaystyle \underset{i}{\Sigma }}\frac{f\left(\left|{\mathbf{r}}_{s+,i}-\mathbf{r}\right|,\left|{\mathbf{r}}_{s-,i}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathit{ax}}{D}_x}}$$

and $${{<t>}_m}^{\mathit{slab}}=\frac{1}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{{\mathit{at}}_m}{D}_m}}{\displaystyle \underset{i}{\Sigma }}\frac{{\mathit{r\text{'}}}_{+,i}{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),{\mathit{r\text{'}}}_{+,i}\right)-{\mathit{r\text{'}}}_{-,i}{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),{\mathit{r\text{'}}}_{-,i}\right)}{{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),{\mathit{r\text{'}}}_{+,i}\right)-{\mathrm{BOY\; WUT}}^{\infty }\left(\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right),{\mathit{r\text{'}}}_{-,i}\right)}={\displaystyle \underset{i}{\Sigma }}\frac{f\left(\left|{\mathbf{r}}_{d+,i}-\mathbf{r}\right|,\left|{\mathbf{r}}_{d-,i}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)}{2{\mathrm{vs}}_{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{{\mathit{at}}_m}{D}_m}}$$

Gold, $${<t>}^{\mathit{slab}}1/2\infty ={<t>}^{\mathit{slab}}x,1/2\infty +{<t>}^{\mathit{slab}}m,1/2\infty +\mathrm{\tau }$$

Is : $$\begin{array}{|cc|}\hline {<t>}_{\mathit{slab}}={\displaystyle \underset{i}{\Sigma }}\left[\frac{\mathit{f}\left(\left|{\mathbf{r}}_{\mathbf{s}i+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}i-}-\mathbf{r}\right|,k_x\left(0\right)\right)}{{\mathrm{\nu }}_x}+\frac{\mathit{f}\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}i+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}i-}\right|,{\mathrm{<figure-callout\; id="0"\; label="k\; m"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k\; </figure-callout>}}_m\left(0\right)\right)}{{\mathrm{\nu }}_m}\right]+\mathrm{\tau }& \\ \hline\end{array}$$

In practice, however, for media of sufficient thickness, the sum converges fairly rapidly, for example when i is equal to or close to 10, for example i = 8, 9, 11 or 12.

The described surface is still a 3D surface. So, again, a fluorophore is located by looking for intersections between 3 surfaces, each obtained for a different point pair (source position, measurement position).

To find the 3D position of the fluorophore, three measurements are made with three different positions sources and 3 different detector positions, as already explained above.

We shall now deal with the case of a medium of any form: this medium is limited by an envelope, but the form of the latter is a priori indeterminate. The most general method for dealing with this case is as follows.

We proceed first to a mesh of the medium.

The equations are then solved numerically (for example by the finite volume method, or the finite element method, etc.).

où : $$F\left(t\right)=\frac{e^{{}^{-t}{/}_{\tau }}}{\mathrm{\tau }},$$

$$\frac{1}{c_n}\frac{\partial {\mathrm{\phi }}_x\left(\mathbf{r},t\right)}{\partial t}+\overrightarrow{\nabla }\mathrm{.}\left(-D_x\left(\mathbf{r}\right)\nabla {\mathrm{\phi }}_x\left(\mathbf{r},t\right)\right)+{\mu }_{\mathit{ax}}\left(\mathbf{r}\right){\mathrm{\phi }}_x\left(\mathbf{r},t\right)=\mathrm{\delta }\left({\mathbf{r}}_s-\mathrm{r}\right)\mathrm{\delta }\left(t\right),$$

avec la condition aux limites (3) ci-dessus.
Et : $$\frac{1}{c_n}\frac{\partial {\mathrm{\phi }}_m\left(\mathbf{r},t\right)}{\partial t}+\overrightarrow{\nabla }\mathrm{.}\left(-D_m\left(\mathbf{r}\right)\nabla {\mathrm{\phi }}_m\left(\mathbf{r},t\right)\right)+{\mu }_{\mathit{am}}\left(\mathbf{r}\right){\mathrm{\phi }}_m\left(\mathbf{r},t\right)=\mathrm{\delta }\left({\mathbf{r}}_d-\mathrm{r}\right)\mathrm{\delta }\left(t\right),$$

avec la condition aux limites (4) ci-dessus.The fluorescence signal can be expressed as follows: $${\mathrm{\Phi }}_m\left({\mathbf{r}}_s,{\mathbf{r}}_d,\mathbf{r},t\right)\alpha {\mathrm{\phi }}_x\left({\mathbf{r}}_s,\mathbf{r},t\right)*F\left(t\right)*{\mathrm{\phi }}_m\left(\mathbf{r},{\mathbf{r}}_d,t\right),$$

or : $$F\left(t\right)=\frac{e^{{}^{-t}{/}_{\tau }}}{\mathrm{\tau }},$$

$$\frac{1}{{\mathrm{vs}}_{\mathrm{not}}}\frac{\partial {\mathrm{\phi }}_x\left(\mathbf{r},t\right)}{\partial t}+\overrightarrow{\nabla }\mathrm{.}\left(-D_x\left(\mathbf{r}\right)\nabla {\mathrm{\phi }}_x\left(\mathbf{r},t\right)\right)+{\mu }_{\mathit{ax}}\left(\mathbf{r}\right){\mathrm{\phi }}_x\left(\mathbf{r},t\right)=\mathrm{\delta }\left({\mathbf{r}}_s-\mathrm{r}\right)\mathrm{\delta }\left(t\right),$$

with the boundary condition (3) above.
And: $$\frac{1}{{\mathrm{vs}}_{\mathrm{not}}}\frac{\partial {\mathrm{\phi }}_m\left(\mathbf{r},t\right)}{\partial t}+\overrightarrow{\nabla }\mathrm{.}\left(-D_m\left(\mathbf{r}\right)\nabla {\mathrm{\phi }}_m\left(\mathbf{r},t\right)\right)+{\mu }_{\mathit{am}}\left(\mathbf{r}\right){\mathrm{\phi }}_m\left(\mathbf{r},t\right)=\mathrm{\delta }\left({\mathbf{r}}_d-\mathrm{r}\right)\mathrm{\delta }\left(t\right),$$

with the boundary condition (4) above.

We thus obtain, in every point r, the value of the functions φ _x ( r _s , r , t ) and φ _m ( r , r _d , t ), which we can calculate the values of the moments, also at any point r , by applying the relation (11) above.

où les fonctions f' sont donc calculées numériquement. On cherche ensuite les points r tels que cette relation soit vraie.So we have the relationship: $$\begin{array}{|c|}\hline \begin{array}{cc}{<t>}_{\mathit{measured\; signal}}\hfill & ={<t>}_x+{<t>}_m+{<t>}_{\mathit{fluorescent\; decline}}\hfill \\ & =\mathit{f\text{'}}\left(\left|{\mathbf{r}}_{\mathbf{s}}-\mathbf{r}\right|\right)+\mathit{f\text{'}}\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}}\right|\right)+\mathrm{\tau }\hfill \end{array}\\ \hline\end{array},$$

where the functions f 'are thus calculated numerically. We then look for the points r such that this relation is true.

avec :

• f_x'(Dx, µax, |rs-r|) : solution de l'équation d'excitation
• f_m' (Dm, µam, |r-rd|) : solution de l'équation d'émission.

We can include the parameters D and μ in the function f 'and thus obtain the expression: $$<\mathrm{t}>\mathrm{measured\; signal}={\mathrm{f}}_{\mathrm{x}}\mathrm{\text{'}}\left(\mathrm{dx},\mathrm{\mu ax},\mathrm{abs}\left(\mathrm{rs}-\mathrm{r}\right)\right)+{\mathrm{f}}_{\mathrm{m}}\mathrm{\text{'}}\left(\mathrm{dm},\mathrm{\mu am},\left|\mathrm{r}-\mathrm{rd}\right|\right)+\mathrm{\tau },$$

with:

• f _x '(Dx, μax, | rs-r |): solution of the excitation equation
• f _m '(Dm, μam, | r-rd |): solution of the emission equation.

By numerical resolution, we find points r that satisfy this equation and we can define a surface passing through all these points, for example by interpolation from one point to another. This surface is not necessarily an ellipsoid.

Finally, we can treat the case where the medium is not homogeneous from the point of view of its optical properties, which then become dependent on r. In this case, as in the case of a homogeneous medium of any shape, one proceeds to the numerical resolution of 3 equations, and one identifies the pair of points satisfying the three equations.

Et on en déduit ensuite $<t>=i\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}|\mathrm{\omega }=0\times \frac{1}{\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)|\mathrm{\omega }=0}$

The expression of Φ (ω) for this medium is then numerically determined, satisfying the written diffusion equation in the frequency domain: $$\overrightarrow{\nabla }\mathrm{.}\left(-D\left(\mathbf{r}\right)\nabla \tilde{\mathrm{\Phi }}\left(\mathbf{r},\mathrm{\omega }\right)\right)+\left({\mu }_{\mathrm{at}}\left(\mathbf{r}\right)-\frac{i\mathrm{\omega }}{{\mathrm{vs}}_{\mathrm{not}}}\right)\tilde{\mathrm{\Phi }}\left(\mathbf{r},\mathrm{\omega }\right)=S\left({\mathbf{r}}_s\right)$$

And then we deduce $<t>=i\frac{\partial \tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)}{\partial \mathrm{\omega }}|\mathrm{\omega }=0\times \frac{1}{\tilde{\mathrm{\Phi }}\left(\mathrm{\omega }\right)|\mathrm{\omega }=0}$

Whatever the proposed geometry, it can be free of the dependence on the lifetime τ, by performing a measurement with a 4 ^th 4 ^th torque positions (source position, detector position) different from each of the 3 previous ones. For additional couple of positions (source, detector), one proceeds to the establishment of a TPSF or a histogram as explained above. One then obtains the equation of a ^4th surface.

As indicated above in each of the treated cases, a calculation of the mean time from each fluorescence signal is actually obtained.

It is described in the document EP 1 884 765 that the calculation of a variable independent of τ can be carried out by making the difference between the mean time calculated for each fluorescence signal and the average time calculated for a particular fluorescence signal, for example the particular fluorescence signal is that having a minimum average time or having a minimum calculated average time.

Among the 4 measurements made for the 4 pairs of positions, we can select the one that corresponds to a minimum measured average time. The surface equation corresponding to this minimum measure will be subtracted from each of the other 3 equations.

We will then obtain 3 new equations of 3 new surfaces, which may not be ellipsoids, but at the intersection of which the fluorophore will be located, or the fluorophore will be located in a volume containing this intersection.

Whatever the geometry envisaged, the calculation of the intersection of 3 ellipsoids, or 3 surfaces, defined or defined, in space by 3 different equations is carried out using a digital processing implemented by a calculator, for example the means 24 of the Figures 1A and 1B . This treatment leads to a localization of the intersection of the 3 surfaces with a certain precision. In other words, the resolution of the system of equations constituted by the definitions of the 3 surfaces is carried out with a certain precision, which will give an approximate solution: the fluorophore is then not located exactly at the intersection of 3 surfaces, but at the neighborhood of this intersection.

After finding a solution with a first precision, we can perform the calculation again to find another solution with a second precision, different from the first.

Each surface can itself be defined with some precision. This no longer defines a surface in the strict sense, but a volume that leans on said surface. In the case of an ellipsoid, it may be an ellipsoidal crown. This volume is delimited by two surfaces, substantially parallel to the surface defined strictly by the corresponding equation and close to it, the proximity being defined by the precision associated with the surface, which should preferably be less than 25%, and preferably less than ± 10%. In other words, it is then not the intersection of 3 surfaces, but 3 volumes. The result is not a single point, but the identification of a volume, generally small enough to be compatible with location with some uncertainty; this volume contains the intersection of the 3 surfaces as defined strictly by the 3 initial equations.

In general, we can distinguish different types of analysis.

First there is the case of an invasive analysis for which source and detector are placed inside the diffusing medium. If the depth of the source and the detector with respect to the middle boundary is sufficient, we shall consider that we are in the conditions of an infinite medium (case of the figure 10A ).

A first case of non-invasive analysis is that for which source and detector are placed in contact with a boundary of the diffusing medium (case of the figure 10B ).

A second non-invasive analysis case is one in which the source and detector are not in physical contact with a boundary 21 of the scattering medium 20, but are in optical contact with the medium. In this case, the source is for example a laser 8 focused on the surface 21 of the medium and the detector is a photodetector 4 in optical contact with the surface of the medium (case of the Figure 10C ).

• la géométrie semi-infinie,
• ou la géométrie de type tranche,
• ou la géométrie de forme quelconque.

The first non-invasive case can be combined with the second non-invasive case, source 8 being located at a distance (like the laser 8 of the figure 10D ) and the detection taking place in contact, for example using an optical fiber 12 which causes the radiation to be detected to the detector 4 (case of the figure 10D ). Furthermore, the excitation can be carried out in contact, for example by means of optical fibers. Generally, in non-invasive mode, we do not consider that we are in an infinite geometry, and we then choose another type of geometry:

• semi-infinite geometry,
• or the slice type geometry,
• or geometry of any shape.

The interest of approaching the infinite, semi-infinite or slice-type geometry is to allow to use an analytic relation linking a measured quantity (for example the average arrival time of the photons) to the respective distances source-fluorophore ( or aggregation or local accumulation of fluorophore) and detector -flurophore (or aggregation or local accumulation of fluorophore). Thus, for 3 different acquisitions, that is to say performed with 3 different pairs (source - detector), the problem is to determine the solutions satisfying three analytical equations, which can be undertaken quickly using current calculation means. On the other hand, as described below, the fact of placing oneself in any geometry amounts to solving the problem numerically, to the detriment of the calculation time.

In the various cases explained above in connection with the Figures 10A-10D 20 denotes a studied medium, the limit of which is the wall 21. This medium may be for example an organ of an animal or of a human being, for example the brain, or a breast (as a examples of fluorophores for these different media include indocyanine green, or fluorescein).

In these figures, the point F denotes a fluorophore or an aggregation of fluorophores. But it is recalled that it may also be an absorber (or a fluorophore re-emitting at the same energy as excitation, and with a duration τ = 0) or an aggregation of absorbers. In this case, the medium 20 may still be an organ, zone F identifying a cancerous tumor, visible by the simple fact that it has a greater attenuation coefficient than the surrounding healthy tissue.

Examples will now be described.

Examples : - Example I-

This example is a two-dimensional calculation. It does not concern a real measure, but illustrates the method in a theoretical case, in a plan.

The medium is assumed to be infinite, 2-dimensional, with the following optical properties.

The reduced scattering coefficient μ _'s = 10 cm ^-1 which leads to: D = 1 / 3μ' _s (this definition of D is the most common, but there are other definitions, for example D = 1 / 3 (μ ' _s + μ _a ), which does not change anything if μ _a <<μ' _s which generally defines the validity framework of the diffusion approximation).

The absorption coefficient is μ _a = 0.1cm ^-1 .

It is further assumed that the optical properties are the same at both wavelengths, refractive index n = 1.0, which gives a propagation speed in the medium c _n = 3.10 ⁸ / n.

This results in an apparent speed: $$v_{\mathit{app}}=2\frac{\mathrm{vs}}{\mathrm{not}}\sqrt{{\mathrm{\mu }}_{\mathrm{at}}D}=3.4469\mathrm{x}{10}^{7}m\mathrm{.}{\mathrm{s}}^{-1}\mathrm{.}$$

The chosen fluorophore has a life time τ = 1.5 ns, a standard order of magnitude of the fluorophores that are used in optical molecular imaging.

• une première mesure (<t>_∞-τ)×ν _app = 4.7000 cm, avec la source positionnée en surface du milieu en (0.0, 0.0) et un détecteur positionné en (1.0 cm,0.0). Cela signifie que le fluorophore est placé sur la demi-ellipse d'équation: $${\mathrm{\xi }}_1:\sqrt{{\left(x-x_{s1}\right)}^2+{\left(y-y_{s1}\right)}^2}+\sqrt{{\left(x-x_{d1}\right)}^2+{\left(y-y_{d1}\right)}^2}=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; dʹordre}1\mathit{mesuré}=m1}{\underbrace{{\langle t\rangle }_{\mathit{mes}1}}}-\mathrm{\tau }\right)=4.7$$
  
• une seconde mesure (<t>_∞-τ)×ν _app = 5.1000cm, avec la source positionnée en surface du milieu en (0.5 cm, 0.0) et un détecteur en (1.5 cm,0.0). Cela signifie que le fluorophore est placé sur la demi-ellipse d'équation $${\mathrm{\xi }}_2:\sqrt{{\left(x-x_{s2}\right)}^2+{\left(y-y_{s2}\right)}^2}+\sqrt{{\left(x-x_{d2}\right)}^2+{\left(y-y_{d2}\right)}^2}=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; dʹordre}1\mathit{mesuré}=m1}{\underbrace{{\langle t\rangle }_{\mathit{mes}2}}}-\mathrm{\tau }\right)=5.1$$
  

We suppose that we realize:

• a first measurement ( <t> _∞ -τ ) × ν _app = 4.7000 cm, with the source positioned on the surface of the medium in (0.0, 0.0) and a detector positioned in (1.0 cm, 0.0). This means that the fluorophore is placed on the half-ellipse of equation: $${\mathrm{\xi }}_1:\sqrt{{\left(x-x_{s1}\right)}^2+{\left(\mathrm{there}-{\mathrm{there}}_{s1}\right)}^2}+\sqrt{{\left(x-x_{d1}\right)}^2+{\left(\mathrm{there}-{\mathrm{there}}_{d1}\right)}^2}=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=4.7$$
  
• a second measurement ( <t> _∞ - τ) × ν _app = 5.1000cm, with the source positioned on the surface of the medium in (0.5 cm, 0.0) and a detector in (1.5 cm, 0.0). This means that the fluorophore is placed on the equation half ellipse $${\mathrm{\xi }}_2:\sqrt{{\left(x-x_{s2}\right)}^2+{\left(\mathrm{there}-{\mathrm{there}}_{s2}\right)}^2}+\sqrt{{\left(x-x_{d2}\right)}^2+{\left(\mathrm{there}-{\mathrm{there}}_{d2}\right)}^2}=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}2}}}-\mathrm{\tau }\right)=5.1$$
  

The figure 3 represents the intersection (zone III) of the two ellipses I and II thus defined, this intersection gives the position of the fluorophore. In two dimensions, the method according to the invention therefore gives satisfactory results. Examples in 3 dimensions will now be given.

Example II-

The experimental setup ( Figure 4A ) is substantially that of the Figure 1A with a laser diode 8 of emission wavelength at about 635 nm (H & B laser diode) as the excitation source and a photomultiplier 4 coupled to a TCSPC card.

• µa = 0.01 cm^-1,
• µ's = 8.7 cm^-1

The tank 21 is filled with a liquid diffusing medium 20 composed of water and Acronal® beads (acrylic resin, BASF), which leads to the following optical parameters:

• μa = 0.01 cm ^-1 ,
• μ's = 8.7 cm ^-1

It is considered that the optical properties are identical to the excitation wavelength and the emission wavelength, because of the proximity of the wavelengths of excitation and diffusion, sufficient for consider these coefficients constant over this wavelength range.

The fluorescent medium in question is Cy5 of concentration 10 .mu.mol. The lifetime τ is 0.96 ns, and the refractive index n ₀ of the medium is 1.33.

The Figure 4A shows in more detail the positioning of the sample in the tank 21. The latter is placed in a glass capillary tube 23, of internal diameter 1 mm, over a height of about 2 mm.

This figure also shows the portion of the fibers 10 and 12 which are immersed in the examined medium. The end of each fiber is located 3.5 cm below the level of the liquid, which is sufficient to ensure the condition of infinite medium. The power measured at the output of the excitation fiber 10 is 100 μW.

Also shown in this figure are the respective positions of the sample and fiber ends.

From the point of view of modeling, the source is not punctual, everything happens as if it became punctual at a distance of a free travel means of transport is at a distance of 1 / μ's (which is the case in the envisaged configuration, since the distance between the fiber ends and the sample is 0.5 cm, while 1 / μ's <0.12 cm <0.5 cm).

5 measurements are made, with source and detector positions shown below.

• S1 : r_s1=[0.5, 0, -3.5],
• S2 : r_s2=[0, 0.25, -3.5],
• S3 : r_s3= [1, 0.5, -3.5],
• S4 : r_s4= [0.5, 0.5, -3.5];
• S5 : r_s5= [0.5, 0.75, -3.5];

We have selected the following 5 positions Si (i = 1, ... 5) of the source (in fact: the end of the fiber 10, all the coordinates are in cm):

• S1: r _s1 = [0.5, 0, -3.5],
• S2: r _s2 = [0, 0.25, -3.5],
• S3: r _s3 = [1, 0.5, -3.5],
• S4: r _s4 = [0.5, 0.5, -3.5];
• S5: r _s5 = [0.5, 0.75, -3.5];

• D1: r_d1=[-0.5, 0, -3.5],
• D2: r_d2= [0, -0.25, -3.5],
• D3: r_d3= [0, 0.5, -3.5];
• D4: r_d4= [0.5, 0, -3.5] ;
• D5: r_d5= [0.5, 0.25, -3.5]

And the following 5 positions Di (i = 1, ... 5) of the detector (in fact: the end of the fiber 12, all coordinates also in cm):

• D1: r _d1 = [- 0.5, 0, -3.5],
• D2: r _d2 = [0, -0.25, -3.5],
• D3: r _d3 = [0, 0.5, -3.5];
• D4: r _d4 = [0.5, 0, -3.5];
• D5: r _d5 = [0.5, 0.25, -3.5]

For the Si position of the source, the detector is in position Di.

The Figure 4B gives each of the positions Si and Di (i = 1, ... 5) in the X plane, Y (Z is fixed at -3.5 cm). The distances di calculated between Si and Di are: d1 = 2.27 cm; d2 = 2.01 cm; d3 = 2.73 cm; d4 = 2.39 cm; d5 = 2,53cm.

The figure 4C gives the fluorescence curve obtained for each pair of positions (Si, Di). The curves were previously processed as explained in A.Laidevant et al., Applied Optics, 45, 19, 4756 (2006) ).

• Pour (S1, D1): 2, 2700 cm ;
• Pour (S2, D2): 2,0059 cm ;
• Pour (S3, D3): 2,7284 cm ;
• Pour (S4, D4): 2,3931 cm ;
• Pour (S5, D5): 2,5258 cm.

The measured values of v _app (<t> _mes1 ) are as follows (remember that <t> _mes1 is the measured moment of order 1):

• For (S1, D1): 2.2700 cm;
• For (S2, D2): 2.0059 cm;
• For (S3, D3): 2.7284 cm;
• For (S4, D4): 2.3931 cm;
• For (S5, D5): 2.5258 cm.

From the measurements taken for 3 of the 5 pairs of points ((S1, D1), (S3, D3) and (S4, D4)), three three-dimensional surfaces, in fact three ellipsoids, were calculated in accordance with the teaching of the 'invention.

The intersection of these three surfaces has also been calculated.

• En x : de 50 points entre x = -0,5 cm et x=2 cm;
• En y _: de 30 points entre y = -0,5 cm et y=1 cm;
• En z : de 50 points entre z = -6 cm et z=-3,5 cm.

The 3 surfaces were developed considering a mesh:

• In x: 50 points between x = -0.5 cm and x = 2 cm;
• In y _: 30 points between y = -0.5 cm and y = 1 cm;
• In z: 50 points between z = -6 cm and z = -3.5 cm.

• -0,041<x₁<0,062;
• 0,12<y₁<0,17
• -0,91<z₁<-3,81.

The fluorophore has been located in the following position (x ₁ , y ₁ , z ₁ ):

• -0.041 <x ₁ <0.062;
• 0.12 <y ₁ <0.17
• -0.91 <z ₁ <-3.81.

Remember that the real position is (0,0, -4). The Figures 4D and 4E represent in each of the planes XY and ZY (respectively respectively in top view and side view of the tank 21) the measured position p ₁ and the calculated position P ₀ , with a slight shift between these two positions. The figure 4F represents these two positions in three dimensions, with, also, a slight difference between these two positions. Despite this discrepancy, it is found that the information obtained by the calculation is quite satisfactory for a good number of cases, for example for an approximate location in an organ of the human body.

• durée du calcul des distances di pour les trois couples (Si, Di) sélectionnés: 2,6 s
• durée du calcul des équations des 3 ellipsoïdes (ξ1, ξ2, et ξ3): 0,12 s
• durée du calcul de la détermination de l'intersection : 0,04 s.

The time required for these calculations, for the mesh chosen (with the selected mesh) with software "Matlab", on Intel Core2 processor, at 2.13GHz, with 1GB of RAM is less than 3 seconds:

• duration of the calculation of the distances di for the three couples (Si, Di) selected: 2.6 s
• calculation time of the equations of the 3 ellipsoids (ξ1, ξ2, and ξ3): 0.12 s
• calculation duration of the intersection determination: 0.04 s.

The total computation time with standard computer hardware is quite compatible with a measurement on or in an organ of the human body, for example during a surgical procedure or in a unit of analysis.

The calculation was then redone, taking into account all the 5 measurements: we thus calculated the intersection of the 5 surfaces.

The fluorophore was then located in position (x ₂ , y ₂ , z ₂ ) = (0.06, 0.12, 3.91). This new position is quite close to the previous one: the location made with only three surfaces is therefore sufficient.

Example III

This example is in three dimensions, and in semi-infinite medium.

We consider 3 pairs of positions of the source and detector (r _s , r _d ).

For example, we have the following 3 positions of the source (all coordinates are in cm): r _s1 = [0.0, 0, 0], r _s2 = [0.5, 0, 0], r _s3 = [0.75, - 1, 0];

• r_d1=[1.0, 0, 0], r_d2= [1.5, 0, 0], r_d3= [0.75, 0, 5, 0]

And the following 3 positions of the detector (all coordinates also in cm):

• r _d1 = [1.0, 0, 0], r _d2 = [1.5, 0, 0], r _d3 = [0.75, 0, 5, 0]

These 3 couples are represented in Figure 5A .

The optical properties are the same as in the previous example.

• une première mesure (<t>_∞-τ)×ν _app = 4.6000 cm, pour le couple de positions (r_s1, r_d1),
• une deuxième mesure (<t>_∞-τ)×ν _app = 5.6000 cm, pour le couple de positions (r_s2, r_d2),
• une troisième mesure (<t>_∞-τ)×ν _app = 4.6000 cm, pour le couple de positions (r_s3, r_d3).

We suppose that we realize:

• a first measurement ( <t> _∞ -τ ) × ν _app = 4.6000 cm, for the pair of positions (r _s1 , r _d1 ),
• a second measurement (<t> _∞ - τ) × ν _app = 5.6000 cm, for the pair of positions (r _s2 , r _d2 ),
• a third measurement ( <t> _∞ -τ ) × ν _app = 4.6000 cm, for the pair of positions (r _s3 , r _d3 ).

$${\mathrm{\xi }}_3:f\left(\left|{\mathbf{r}}_{\mathbf{s}3+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}3-}-\mathbf{r}\right|,k\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3-}\right|,k\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; dʹordre}1\mathit{mesuré}=m1}{\underbrace{{\langle t\rangle }_{\mathit{mes}1}}}-\mathrm{\tau }\right)=5$$

$${\mathrm{\xi }}_1:f\left(\left|{\mathbf{r}}_{\mathbf{s}1+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}1-}-\mathbf{r}\right|,k\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1-}\right|,k\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; dʹordre}1\mathit{mesuré}=m1}{\underbrace{{\langle t\rangle }_{\mathit{mes}1}}}-\mathrm{\tau }\right)=4.6$$

Then the position of the fluorophore belongs to the intersection of the ellipsoids, which have the following equations: $${\mathrm{\xi }}_2:f\left(\left|{\mathbf{r}}_{\mathbf{s}2+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}2-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}2+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}2-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=4.6$$

$${\mathrm{\xi }}_3:f\left(\left|{\mathbf{r}}_{\mathbf{s}3+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}3-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=5$$

$${\mathrm{\xi }}_1:f\left(\left|{\mathbf{r}}_{\mathbf{s}1+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}1-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=4.6$$

On each of Figures 5B , 5C, 5D is shown a portion of each of the corresponding ellipsoids.

The intersection zone of these 3 ellipsoids is identified by zone I in Figure 6A , on which are further represented the three positions S1, S2, S3 of the source and the three positions D1, D2, D3 of the detector used for the three measurements. Zone P is the localization area of the fluorophore.

• En x : de 101 points entre x = -2 et x=3;
• En y : de 101 points entre x = -2 et x=2;
• En z : de 101 points entre x = -3 et x=0;

The figures were developed considering a mesh:

• In x: of 101 points between x = -2 and x = 3;
• In y: of 101 points between x = -2 and x = 2;
• In z: 101 points between x = -3 and x = 0;

The solution of the Figure 6A is an approximate solution, within 10%.

• En figure 6B : solution à 2% près,
• En figure 6C : solution à 1,4% près.

It is possible to impose a greater precision. So, in Figures 6B and 6C , are represented solutions (zone P) more precise:

• In Figure 6B : 2% solution,
• In Figure 6C : solution to 1.4%.

It can be seen that the increase in precision provides little additional information for an approximate location of the fluorophore.

• pour le calcul des moments pour les 3 couples source-détecteur en chaque point du maillage : 5.6835 s,
• pour le calcul des équations des 3 ellipsoïdes (ξ1, ξ2, et ξ3): 2.086609 s,
• pour le calcul de la détermination de l'intersection : 2.253164 s.

It is interesting to evaluate the time required for these calculations, with software "Matlab", on Intel Core2 processor, at 2.13GHz, with 1GB of RAM:

• for the calculation of the moments for the 3 source-detector pairs at each point of the mesh: 5.6835 s,
• for the calculation of the equations of the 3 ellipsoids (ξ1, ξ2, and ξ3): 2.086609 s,
• for the calculation of the determination of the intersection: 2.253164 s.

This results in a total duration of about 10 s for the chosen mesh.

This result is a good illustration of the "real time" or fast nature of the process.

In the context of a surgical application, a practitioner may perform an analysis according to the invention even though it is in the presence of a patient. A duration of 10 seconds, or even a slightly longer duration, is entirely compatible with keeping a patient on an operating table or in an analysis room.

Determining the position of the fluorophore depends on its life time. To overcome this, one can consider differential measures, as explained in the patent application. EP 1 884 765 .

• Avec les configurations de sources-détecteurs ci-dessus, en milieu semi-infini, on a les ellipsoïdes suivants : $${\mathrm{\xi }}_1:f\left(\left|{\mathbf{r}}_{\mathbf{s}1+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}1-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; dʹordre}1\mathit{mesuré}=m1}{\underbrace{{\langle t\rangle }_{\mathit{mes}1}}}-\mathrm{\tau }\right)=4.7$$
  
  $${\mathrm{\xi }}_2:f\left(\left|{\mathbf{r}}_{\mathbf{s}2+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}2-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}2+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}2-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; dʹordre}1\mathit{mesuré}=m1}{\underbrace{{\langle t\rangle }_{\mathit{mes}1}}}-\mathrm{\tau }\right)=5$$
  
  $${\mathrm{\xi }}_3:f\left(\left|{\mathbf{r}}_{\mathbf{s}3+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}3-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; dʹordre}1\mathit{mesuré}=m1}{\underbrace{{\langle t\rangle }_{\mathit{mes}1}}}-\mathrm{\tau }\right)=4.6$$
  

For example :

• With the above detector-source configurations, in a semi-infinite environment, we have the following ellipsoids: $${\mathrm{\xi }}_1:f\left(\left|{\mathbf{r}}_{\mathbf{s}1+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}1-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}1-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=4.7$$
  
  $${\mathrm{\xi }}_2:f\left(\left|{\mathbf{r}}_{\mathbf{s}2+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}2-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}2+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}2-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=5$$
  
  $${\mathrm{\xi }}_3:f\left(\left|{\mathbf{r}}_{\mathbf{s}3+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}3-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}3-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=4.6$$
  

The third corresponds to a measurement whose optical path is the shortest of the three. It is this measure which, as explained in the application EP 1 884 765 , can be used as a reference measure.

et $${\mathrm{\xi }}_2-{\mathrm{\xi }}_{\mathrm{3}}=\mathrm{0.4},$$

on obtient des relations indépendantes du temps de vie.If we consider the quantities $${\mathrm{\xi }}_{\mathrm{1}}-{\mathrm{<figure-callout\; id="3"\; label="\xi "\; filenames="imgf0003.png,imgf0008.png"\; state="\{\{state\}\}">\xi </figure-callout>}}_{\mathrm{3}}=\mathrm{0.1}$$

and $${\mathrm{\xi }}_2-{\mathrm{<figure-callout\; id="3"\; label="\xi "\; filenames="imgf0003.png,imgf0008.png"\; state="\{\{state\}\}">\xi </figure-callout>}}_{\mathrm{3}}=\mathrm{0.4},$$

we obtain relations independent of the life time.

These quantities no longer define ellipsoids but more complex surfaces.

The Figures 7A and 7B represent, in two different orientations, the intersection between the two 3D surfaces ξ ₁ -ξ ₃ (zone I in these figures) and ξ ₂ -ξ ₃ (zone II in these figures).

In 3D, a third surface is needed to identify a possible location of the fluorophore.

So we consider a set, not of 3 measures, but of 4 measures.

For example, we have the following 4 positions of the source (all coordinates are in cm): r _s1 = [0.0, 0, 0], r _s2 = [0.5, 0, 0], r _s3 = [0.75, - 1, 0]; r _s4 = [-0.75; -1.5; 0];

• r_d1=[1.0, 0, 0], r_d2= [1.5, 0, 0], r_d3= [0.75, 0,5, 0] ; r_d4= [0.75, 1, 0].

And the following 4 positions of the detector (all coordinates also in cm):

• r _d1 = [1.0, 0, 0], r _d2 = [1.5, 0, 0], r _d3 = [0.75, 0.5, 0]; r _d4 = [0.75, 1, 0].

In addition to the 3 surfaces ξ ₁ , ξ ₂ , ξ ₃ already defined, we thus obtain a surface ξ ₄ (which also defines an ellipsoid): $${\mathrm{\xi }}_4:f\left(\left|{\mathbf{r}}_{\mathbf{s}4+}-\mathbf{r}\right|,\left|{\mathbf{r}}_{\mathbf{s}4-}-\mathbf{r}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)+f\left(\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}4+}\right|,\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{d}4-}\right|,\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}\left(0\right)\right)=v_{\mathit{app}}\times \left(\underset{\mathit{moment\; of\; order}1\mathit{measured}=m1}{\underset{\}}{{<t>}_{\mathit{my}1}}}-\mathrm{\tau }\right)=5.0$$

This surface ξ ₄ will also be normalized. We thus make the difference ξ ₄ -ξ ₃ which defines itself a surface whose intersection with the surfaces ξ ₁ -ξ ₃ and ξ ₂ -ξ ₃ will allow to locate the fluorophore.

The Figure 7C represents the 3 surfaces ξ ₁ -ξ ₃ , (zone I), ξ ₂ -ξ ₃ (zone II) and ξ ₄ -ξ ₃ (zone III), while the Figure 7D represents a section of the set in an XZ plane.

The zone of intersection of the 3 surfaces, and therefore the zone of localization of the fluorophore, is identified by the zone P on this Figure 7D .

Example IV-

This example is in three dimensions, and in "slab" type geometry.

The slice in question is of thickness L = 1.5 cm.

We consider 3 pairs of positions of the source and detector (r _s , r _d ).

For example, we have the following 3 positions of the source (all coordinates are in cm): r _s1 = [0.0, 0, 0], r _s2 = [0.5, 0, 0], r _s3 = [0.75, - 1, 0];

• r_d1=[1.0, 0, 0], r_d2= [1.5, 0, 0], r_d3= [0.75, 0, 5, 0]

And the following 3 positions of the detector (all coordinates also in cm):

• r _d1 = [1.0, 0, 0], r _d2 = [1.5, 0, 0], r _d3 = [0.75, 0, 5, 0]

As for the previous examples, we obtain the definition of three ellipsoids, for the following three values of v _app (<t> _mes ): 1.84; 1.93; 1.81.

On each of Figures 8A, 8B, 8C is shown a portion of each of the corresponding ellipsoids.

In these figures, the slice constituting the medium is delimited by the two parallel planes: Z = 0 and Z = -1.5 cm.

The intersection zone of these 3 ellipsoids is identified by zone I in figure 8D , on which are furthermore represented the three positions of the source and the three positions of the detector used for the three measurements.

In general, an exemplary method according to the invention can be implemented according to the diagram represented in FIG. figure 9 .

The steps of this example, and moreover of any other implementation of a method according to the invention, may be preceded by a step - not shown - of injection into the medium studied of at least one fluorophore or absorber intended to be fixed on an area of interest. In fact, when a fluorophore is injected, many are often injected. We can also assimilate a fluorescent zone to an agglomeration of fluorophores. There is usually not just one fluorescent molecule, but one local concentration of fluorescent molecules. In the same way, when working in absorption, there is not only one absorbing molecule, but a concentration of absorbing molecules. This concentration can be considered as point if the distance separating it from the detector is sufficiently large in front of its dimensions. Typically, a concentration of radius r is considered as point, at a given observation distance d, if d> 10 r. Some absorbers may already be in the medium, for example when it is a zone with a different absorption of its environment.

The radiation emission and detection means are then positioned relative to the medium. One of the configurations already described can be selected. In particular, a non-invasive invasive examination can be performed.

During a first step S1 of this method, input data or parameters are provided: these are the optical properties of the scattering medium, at the two wavelengths of excitation and emission ( in some cases, at these two wavelengths, the optical properties are identical), the geometry of the medium, the positions of the sources and the detector (in the sense already explained above), and the choice of the mesh for the calculation of the surfaces. During a second step (S2), the measurement files (TPSF) for each source-detector pair are read. Preprocessing of the data can be done.

From the data introduced during the first step, an analytic expression of each surface will be able to be determined or calculated for each point of the mesh (step S3).

The data from step S2 will allow a calculation of mean times <t _i > for each of the measurements made (step S4).

Steps S1 and S3 on the one hand, S2 and S34 on the other hand, are represented in figure 9 as substantially performed simultaneously, but it is not necessary, although desirable for the fastest possible process.

From the calculation data of steps S3 and S4, the points of the mesh are searched such that ξ _i = <t _i >, for all i, to within X% (step S5), which means | ξ _i - <t _i > | ≤ X / 100. The user can himself set the desired X precision.

The results (the intersection of the surfaces between it, and possibly each of the surfaces individually) can then be viewed (step S6).

If necessary, the calculations can be reiterated by setting a value of X higher (in which case the precision is reduced) or lower (in which case the precision is increased) to the previous value (step S7).

Claims (15)

1. Method of localising at least one fluorophore (22) or at least one absorber in a scattering medium (20), by means of a device comprising a pulse radiation source (8, 10) suited to emitting an excitation radiation of this fluorophore or this molecule and detection means (4, 12) suited to measuring a fluorescence signal emitted by this fluorophore (22) or an emission signal emitted by this absorber, said device allowing a time resolved detection of these signals, the method being characterized in that it comprises:

   a) for at least 3 different pairs of positions of the radiation source and detection means, at least one excitation by a radiation coming from the radiation source (8), and at least one detection by the means (4) of detecting the fluorescence signal emitted by this fluorophore after this excitation, or of the emission signal emitted by this absorber,

   b) for each of these pairs, the identification of a surface on which the fluorophore or the absorber is situated, or of a volume comprising this surface and in which the fluorophore or the absorber is situated, this identification comprising the calculation of a moment of order n≥1 of said fluorescence signal or of said emission signal or of a frequency function of said fluorescence signal or of said emission signal,

   c) an estimation of the localisation of the fluorophore or the absorber in its scattering medium, by calculation of the intersection of the three surfaces, and possibly of a volume around this intersection.
2. Method according to claim 1, wherein the diffusing medium, surrounding the fluorophore or the absorber is of infinite type or semi-infinite type or forms a slab, limited by two parallel surfaces, or is of any shape, its exterior surface being discretised into a series of planes.
3. Method according to any of claims 1 or 2, wherein the scattering medium is of infinite type, the surfaces then having the shape of ellipsoids.
4. Method according to any of claims 1 to 3, step b) comprising, for each position pair, a calculation of a normalised time moment, of a time function of the fluorescence signal or of the emission signal by the absorber, or of a normalised moment of a frequency function, deduced from said time function of the fluorescence signal or the emission signal by the absorber.
5. Method according to claim 4, the frequency function being deduced from said time function by Fourier transform or by Laplace transform or by Mellin transform.
6. Method according to any of claims 1 to 5, wherein an excitation is carried out by the radiation source (8), and a detection by the detection means (4) for 4 different pairs of positions of the radiation source and detection means.
7. Method according to claim 6, wherein, for the localisation of at least one fluorophore:

   - among the four pairs of points are selected the pair of points, known as fourth pair of points, for which the average arrival time of the photons, from the time of emission by the emitter to the time of reception by the receiver, is the shortest, and, for this pair of points, the equation of a surface on which the fluorophore is situated is calculated,

   - and, for each of the three other pairs of points, a first equation of a first surface on which the fluorophore is situated is calculated, and a difference is taken between this equation and that associated with the fourth pair, to obtain a second equation of a second surface independent of the lifetime of the fluorophore
8. Method according to any of claims 1 to 6, wherein, for the localisation of at least one fluorophore (22), step b) comprises, for each position pair, a calculation of an equation of each surface, independent of the lifetime of the fluorophore.
9. Method according to claim 8, said equation independent of τ resulting from the difference between the average time measured or calculated for each position pair and the average time measured or calculated for a fourth position pair.
10. Method according to claim 9, the fourth position pair being a pair for which the fluorescence signal has a measured average time or a calculated average time less than that of the 3 position pairs.
11. Method according to any of claims 1 to 10, the radiation source being the end of an optic fibre (10) which sends light into the medium and/or and the detection means being the end of an optic fibre (12), which samples a part of the emission light.
12. Device for the localisation of at least one fluorophore or at least one absorber in a scattering medium, comprising:

    - a pulse radiation source (8, 10) suited to emitting an excitation radiation,

    - detection means (4, 12) suited to measuring a fluorescence signal emitted by this fluorophore (22) or an emission signal emitted by this absorber,

    - the device allowing a time resolved detection of these signals,

    this device being characterized in that it comprises:

    calculation means for:

    a) for each pair of points among at least 3 different pairs of positions of the radiation source and detection means, carrying out a calculation or determining, comprising the calculation of a moment of order n > 1 of said fluorescence signal or said emission signal, after an excitation carried out by a radiation coming from the radiation source (8), and a detection by the means (4) of detecting signals emitted by the fluorophore or the absorber after this excitation, of a surface on which the fluorophore or the absorber is situated, or a volume comprising this surface and in which the fluorophore or the absorber is situated,

    b) and for estimating the localisation of the fluorophore or the absorber in its surrounding medium, by calculation of the intersection of the three surfaces or of a volume around this intersection.
13. Device according to claim 12, the calculation means making it possible to calculate, for each position pair, a normalised time moment of a time function, or a normalised moment of a frequency function, deduced from said emission time function.
14. Device according to claim 12 or 13, said detection means comprising TCSPC type means or camera type means.
15. Device according to one of claims 12 to 14, further comprising means of visual or graphic representation of the localisation of the fluorophore or the absorbent molecule.

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